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7.3.6: Applying Volume and Surface Area


Lesson

Let's explore things that are proportional to volume or surface area.

Exercise (PageIndex{1}): You Decide

For each situation, decide if it requires Noah to calculate surface area or volume. Explain your reasoning.

  1. Noah is planning to paint the bird house he built. He is unsure if he has enough paint.
  2. Noah is planning to use a box with a trapezoid base to hold modeling clay. He is unsure if the clay will all fit in the box.

Exercise (PageIndex{2}): Foam Play Structure

At a daycare, Kiran sees children climbing on this foam play structure.

Kiran is thinking about building a structure like this for his younger cousins to play on.

  1. The entire structure is made out of soft foam so the children don’t hurt themselves. How much foam would Kiran need to build this play structure?
  2. The entire structure is covered with vinyl so it is easy to wipe clean. How much vinyl would Kiran need to build this play structure?
  3. The foam costs 0.8¢ per in3. Here is a table that lists the costs for different amounts of vinyl. What is the total cost for all the foam and vinyl needed to build this play structure?
vinyl (in2)cost ($)
(75)(0.45)
(125)(0.75)
Table (PageIndex{1})

Are you ready for more?

When he examines the play structure more closely, Kiran realizes it is really two separate pieces that are next to each other.

  1. How does this affect the amount of foam in the play structure?
  2. How does this affect the amount of vinyl covering the play structure?

Exercise (PageIndex{3}): Filling the Sandbox

The daycare has two sandboxes that are both prisms with regular hexagons as their bases. The smaller sandbox has a base area of 1,146 in2 and is filled 10 inches deep with sand.

  1. It took 14 bags of sand to fill the small sandbox to this depth. What volume of sand comes in one bag? (Round to the nearest whole cubic inch.)
  2. The daycare manager wants to add 3 more inches to the depth of the sand in the small sandbox. How many bags of sand will they need to buy?
  3. The daycare manager also wants to add 3 more inches to the depth of the sand in the large sandbox. The base of the large sandbox is a scaled copy of the base of the small sandbox, with a scale factor of 1.5. How many bags of sand will they need to buy for the large sandbox?
  4. A lawn and garden store is selling 6 bags of sand for $19.50. How much will they spend to buy all the new sand for both sandboxes?

Summary

Suppose we wanted to make a concrete bench like the one shown in this picture. If we know that the finished bench has a volume of 10 ft3 and a surface area of 44 ft2 we can use this information to solve problems about the bench.

For example,

  • How much does the bench weigh?
  • How long does it take to wipe the whole bench clean?
  • How much will the materials cost to build the bench and to paint it?

To figure out how much the bench weighs, we can use its volume, 10 ft3. Concrete weighs about 150 pounds per cubic foot, so this bench weighs about 1,500 pounds, because (10cdot 150=1,500).

To figure out how long it takes to wipe the bench clean, we can use its surface area, 44 ft2. If it takes a person about 2 seconds per square foot to wipe a surface clean, then it would take about 88 seconds to clean this bench, because (44cdot 2=88). It may take a little less than 88 seconds, since the surfaces where the bench is touching the ground do not need to be wiped.

Would you use the volume or the surface area of the bench to calculate the cost of the concrete needed to build this bench? And for the cost of the paint?

Glossary Entries

Definition: Base (of a prism or pyramid)

The word base can also refer to a face of a polyhedron.

A prism has two identical bases that are parallel. A pyramid has one base.

A prism or pyramid is named for the shape of its base.

Definition: Cross Section

A cross section is the new face you see when you slice through a three-dimensional figure.

For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

Definition: Prism

A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

Here are some drawings of prisms.

Definition: Pyramid

A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

Here are some drawings of pyramids.

Definition: Surface Area

The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.

For example, if the faces of a cube each have an area of 9 cm2, then the surface area of the cube is (6cdot 9), or 54 cm2.

Definition: Volume

Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.

For example, the volume of this rectangular prism is 60 units3, because it is composed of 3 layers that are each 20 units3.

Practice

Exercise (PageIndex{4})

A landscape architect is designing a pool that has this top view:

  1. How much water will be needed to fill this pool 4 feet deep?
  2. Before filling up the pool, it gets lined with a plastic liner. How much liner is needed for this pool?
  3. Here are the prices for different amounts of plastic liner. How much will all the plastic liner for the pool cost?
plastic liner (ft2)cost ($)
(25)(3.75)
(50)(7.50)
(75)(11.25)
Table (PageIndex{2})

Exercise (PageIndex{5})

Shade in a base of the trapezoidal prism. (The base is not the same as the bottom.)

  1. Find the area of the base you shaded.
  2. Find the volume of this trapezoidal prism.

(From Unit 7.3.3)

Exercise (PageIndex{6})

For each diagram, decide if (y) is an increase or a decrease of (x). Then determine the percentage that (x) increased or decreased to result in (y).

(From Unit 4.2.4)

Exercise (PageIndex{7})

Noah is visiting his aunt in Texas. He wants to buy a belt buckle whose price is $25. He knows that the sales tax in Texas is 6.25%.

  1. How much will the tax be on the belt buckle?
  2. How much will Noah spend for the belt buckle including the tax?
  3. Write an equation that represents the total cost, (c), of an item whose price is (p).

(From Unit 4.3.1)


Lesson 16

This activity reinforces what students learned in the previous lesson. Students are given two contextual situations and determine if the situation requires surface area or volume to be calculated.

Launch

Arrange students in groups of 2. Give students 1 minute of quiet work time followed by time to discuss their reasoning with a partner. Follow with a whole-class discussion.

For each situation, decide if it requires Noah to calculate surface area or volume. Explain your reasoning.

Noah is planning to paint the bird house he built. He is unsure if he has enough paint.

Noah is planning to use a box with a trapezoid base to hold modeling clay. He is unsure if the clay will all fit in the box.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select students to share their responses. Ask students to describe why the bird house situation calls for surface area and why the clay context calls for volume. To highlight the differences between the two uses of the box, ask:

  • “What are the differences in how Noah is using the boxes in these situations?”
  • “How can you determine if a situation is asking you to calculate surface area or volume?”

The goal is to ensure students understand the differences between situations that require them to calculate surface area and volume.


Surface Area Of Prisms

A prism is a solid that has two parallel faces which are congruent polygons at both ends. These faces form the bases of the prism. A prism is named after the shape of its base. The other faces are in the shape of parallelograms. They are called lateral faces .

The following diagrams show a triangular prism and a rectangular prism.


A right prism is a prism that has its bases perpendicular to its lateral surfaces.

When we cut a prism parallel to the base, we get a cross section of a prism. The cross section is congruent (same size and shape) as the base, as can be seen in the following diagram.


How To Calculate The Surface Area Of A Prism?
The surface area of a prism is the total area of all its external faces.

Step 1: Determine the shape of each face.
Step 2: Calculate the area of each face.
Step 3: Add up all the areas to get the total surface area.

We can also use the formula:
Surface area of prism = 2 × area of base + perimeter of base × height

Example:
Calculate the surface area of the following prism.


Solution:
There are 2 triangles with the base = 4 cm and height = 3 cm.

1 rectangle with length = 7 cm and width = 5 cm
Area = lw = 7 × 5 = 35 cm 2 <

1 rectangle with length = 7 cm and width 3 m
Area = lw = 7 × 3 = 21 cm 2

1 rectangle with length = 7 cm and width 4 m
Area = lw = 7 × 4 = 28 cm 2

The total surface area is 12 + 35 + 21 + 28 = 96 cm 2

We can also use the formula:
Surface area of prism = 2 × area of base + perimeter of base × height
= 2 × 6 + (3 + 4 + 5) × 7 = 96 cm 2

Example:
The diagram shows a prism whose base is a trapezoid. The surface area of the trapezoidal prism is 72 cm 2 . Find the value of x.

Solution:
There are 2 rectangles with length = 5 cm and width = 3 cm
Area = 2 × 5 × 3 = 30 cm 2

There is one rectangle with length = 5 cm and width = 4 cm
Area = 5 × 4 = 20 cm 2

There is one rectangle with length = 5 cm and width = 2 cm
Area = 5 × 2 = 10 cm 2

There are two trapezoids.
Area = cm 2 = 6x cm 2

Sum of area
30 + 20 + 10 + 6x = 72
60 + 6x = 72
x = 2

How To Find The Surface Area Of Different Types Of Prisms
This video shows how to find the surface area of prisms: cuboid (or rectangular prism), triangular prism, trapezoidal prism.

How To Find The Surface Area Of A Rectangular Prism?

How To Find The Surface Area Of A Triangular Prism Using The Formula SA = ab+(s1+s2+s3)h?

How To Find The Surface Area Of A Pentagonal Prism?

How To Find The Surface Area Of A Hexagonal Prism?

How To Find The Surface Area Of A Octagonal Prism?

Word Problems About Prisms

How to find the surface area of prisms and cylinders using a given formula? How to solve word problems and composite figures?

Problem: A treasure chest is a composite figure. If you were to paint the surface area, how many square feet would you paint? Round your answer to the nearest square feet.

Surface Area Of Prisms Using Nets

This video shows how to find the surface area of a cube, rectangular prism and triangular prism using nets.

How To Find The Surface Area Of A Hexagonal Prism Using A Net?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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Lesson 16

In this second lesson on applying surface area and volume to solve problems, students solve more complex real-word problems that require them to choose which of the two quantities is appropriate for solving the problem, or whether both are appropriate for different aspects of the problem. They use previous work on ratios and proportional relationships, thus consolidating their knowledge and skill in that area. When students bring together knowledge of different areas of mathematics to solve a complex problem, they are engaging in MP4.

Learning Goals

Let's explore things that are proportional to volume or surface area.

Learning Targets

CCSS Standards

Glossary Entries

The word base can also refer to a face of a polyhedron.

A prism has two identical bases that are parallel. A pyramid has one base.

A prism or pyramid is named for the shape of its base.

Expand Image

Description: <p>The figure on the left is labeled pentagonal prism. There are two identical pentagons on the top and bottom. Each vertex of a pentagon is connected by a vertical segment to the corresponding vertex of the other pentagons. The pengatons are each shaded, with the word base pointing to each. The figure on the right is labeled hexagonal pyramid. There is a hexagon on the bottom shaded green. From a point above the hexagon extend 6 segments, each connected to a vertex of the hexagon.</p>

A cross section is the new face you see when you slice through a three-dimensional figure.

For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.

A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

Here are some drawings of prisms.

Expand Image

A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

Here are some drawings of pyramids.

Expand Image

The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.

For example, if the faces of a cube each have an area of 9 cm 2 , then the surface area of the cube is (6 oldcdot 9) , or 54 cm 2 .

Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.

For example, the volume of this rectangular prism is 60 units 3 , because it is composed of 3 layers that are each 20 units 3 .

Expand Image

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IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

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58 Solve Geometry Applications: Volume and Surface Area

  1. Evaluate when
    If you missed this problem, review (Figure).
  2. Evaluate when
    If you missed this problem, review (Figure).
  3. Find the area of a circle with radius
    If you missed this problem, review (Figure).

In this section, we will finish our study of geometry applications. We find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.

  1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
  2. Identify what you are looking for.
  3. Name what you are looking for. Choose a variable to represent that quantity.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.

Find Volume and Surface Area of Rectangular Solids

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See (Figure)). The amount of paint needed to cover the outside of each box is the surface area , a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

Each crate is in the shape of a rectangular solid . Its dimensions are the length, width, and height. The rectangular solid shown in (Figure) has length units, width units, and height units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This by by rectangular solid has cubic units.

Altogether there are cubic units. Notice that is the

The volume, of any rectangular solid is the product of the length, width, and height.

We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, is equal to

We can substitute for in the volume formula to get another form of the volume formula.

We now have another version of the volume formula for rectangular solids. Let’s see how this works with the rectangular solid we started with. See (Figure).

To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

Notice for each of the three faces you see, there is an identical opposite face that does not show.

The surface area of the rectangular solid shown in (Figure) is square units.

In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see (Figure)). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

For each face of the rectangular solid facing you, there is another face on the opposite side. There are faces in all.

For a rectangular solid with length width and height

For a rectangular solid with length cm, height cm, and width cm, find the ⓐ volume and ⓑ surface area.

Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

Step 1. Read the problem. Draw the figure and
label it with the given information.
Step 2. Identify what you are looking for. the volume of the rectangular solid
Step 3. Name. Choose a variable to represent it. Let = volume
Step 4. Translate.
Write the appropriate formula.
Substitute.


Step 5. Solve the equation.
Step 6. Check
We leave it to you to check your calculations.
Step 7. Answer the question. The volume is cubic centimeters.
Step 2. Identify what you are looking for. the surface area of the solid
Step 3. Name. Choose a variable to represent it. Let = surface area
Step 4. Translate.
Write the appropriate formula.
Substitute.


Step 5. Solve the equation.
Step 6. Check: Double-check with a calculator.
Step 7. Answer the question. The surface area is 1,034 square centimeters.

Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length feet, width feet, and height feet.

Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length feet, width feet, and height feet.

A rectangular crate has a length of inches, width of inches, and height of inches. Find its ⓐ volume and ⓑ surface area.

Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

Step 1. Read the problem. Draw the figure and
label it with the given information.
Step 2. Identify what you are looking for. the volume of the crate
Step 3. Name. Choose a variable to represent it. let = volume
Step 4. Translate.
Write the appropriate formula.
Substitute.


Step 5. Solve the equation.
Step 6. Check: Double check your math.
Step 7. Answer the question. The volume is 15,000 cubic inches.
Step 2. Identify what you are looking for. the surface area of the crate
Step 3. Name. Choose a variable to represent it. let = surface area
Step 4. Translate.
Write the appropriate formula.
Substitute.


Step 5. Solve the equation.
Step 6. Check: Check it yourself!
Step 7. Answer the question. The surface area is 3,700 square inches.

A rectangular box has length feet, width feet, and height feet. Find its ⓐ volume and ⓑ surface area.

A rectangular suitcase has length inches, width inches, and height inches. Find its ⓐ volume and ⓑ surface area.

Volume and Surface Area of a Cube

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

So for a cube, the formulas for volume and surface area are and

For any cube with sides of length

A cube is inches on each side. Find its ⓐ volume and ⓑ surface area.

Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

Step 1. Read the problem. Draw the figure and
label it with the given information.
Step 2. Identify what you are looking for. the volume of the cube
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.

Step 5. Solve. Substitute and solve.
Step 6. Check: Check your work.
Step 7. Answer the question. The volume is 15.625 cubic inches.
Step 2. Identify what you are looking for. the surface area of the cube
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.

Step 5. Solve. Substitute and solve.
Step 6. Check: The check is left to you.
Step 7. Answer the question. The surface area is 37.5 square inches.

For a cube with side 4.5 meters, find the ⓐ volume and ⓑ surface area of the cube.

For a cube with side 7.3 yards, find the ⓐ volume and ⓑ surface area of the cube.

A notepad cube measures inches on each side. Find its ⓐ volume and ⓑ surface area.

Step 1. Read the problem. Draw the figure and
label it with the given information.
Step 2. Identify what you are looking for. the volume of the cube
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.

Step 5. Solve the equation.
Step 6. Check: Check that you did the calculations
correctly.
Step 7. Answer the question. The volume is 8 cubic inches.
Step 2. Identify what you are looking for. the surface area of the cube
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.

Step 5. Solve the equation.
Step 6. Check: The check is left to you.
Step 7. Answer the question. The surface area is 24 square inches.

A packing box is a cube measuring feet on each side. Find its ⓐ volume and ⓑ surface area.

A wall is made up of cube-shaped bricks. Each cube is inches on each side. Find the ⓐ volume and ⓑ surface area of each cube.

Find the Volume and Surface Area of Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate with

For a sphere with radius

A sphere has a radius inches. Find its ⓐ volume and ⓑ surface area.

Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

Step 1. Read the problem. Draw the figure and label
it with the given information.
Step 2. Identify what you are looking for. the volume of the sphere
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.

Step 5. Solve.
Step 6. Check: Double-check your math on a calculator.
Step 7. Answer the question. The volume is approximately 904.32 cubic inches.
Step 2. Identify what you are looking for. the surface area of the cube
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.

Step 5. Solve.
Step 6. Check: Double-check your math on a calculator
Step 7. Answer the question. The surface area is approximately 452.16 square inches.

Find the ⓐ volume and ⓑ surface area of a sphere with radius 3 centimeters.

Find the ⓐ volume and ⓑ surface area of each sphere with a radius of foot

A globe of Earth is in the shape of a sphere with radius centimeters. Find its ⓐ volume and ⓑ surface area. Round the answer to the nearest hundredth.

Step 1. Read the problem. Draw a figure with the
given information and label it.
Step 2. Identify what you are looking for. the volume of the sphere
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for )


Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The volume is approximately 11,488.21 cubic inches.
Step 2. Identify what you are looking for. the surface area of the sphere
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for )


Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The surface area is approximately 2461.76 square inches.

A beach ball is in the shape of a sphere with radius of inches. Find its ⓐ volume and ⓑ surface area.

A Roman statue depicts Atlas holding a globe with radius of feet. Find the ⓐ volume and ⓑ surface area of the globe.

Find the Volume and Surface Area of a Cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder <!– no-selfclose –> is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height />of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, />, will be perpendicular to the bases.

Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid, , can also be used to find the volume of a cylinder.

For the rectangular solid, the area of the base, , is the area of the rectangular base, length × width. For a cylinder, the area of the base, is the area of its circular base, (Figure) compares how the formula is used for rectangular solids and cylinders.

To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See (Figure).

The distance around the edge of the can is the circumference of the cylinder’s base it is also the length of the rectangular label. The height of the cylinder is the width of the rectangular label. So the area of the label can be represented as

To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

The surface area of a cylinder with radius and height is

For a cylinder with radius and height

A cylinder has height centimeters and radius centimeters. Find the ⓐ volume and ⓑ surface area.

Step 1. Read the problem. Draw the figure and label
it with the given information.
Step 2. Identify what you are looking for. the volume of the cylinder
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for )


Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The volume is approximately 141.3 cubic inches.
Step 2. Identify what you are looking for. the surface area of the cylinder
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for )


Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The surface area is approximately 150.72 square inches.

Find the ⓐ volume and ⓑ surface area of the cylinder with radius 4 cm and height 7cm.

Find the ⓐ volume and ⓑ surface area of the cylinder with given radius 2 ft and height 8 ft.

Find the ⓐ volume and ⓑ surface area of a can of soda. The radius of the base is centimeters and the height is centimeters. Assume the can is shaped exactly like a cylinder.

Step 1. Read the problem. Draw the figure and
label it with the given information.
Step 2. Identify what you are looking for. the volume of the cylinder
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for )


Step 5. Solve.
Step 6. Check: We leave it to you to check.
Step 7. Answer the question. The volume is approximately 653.12 cubic centimeters.
Step 2. Identify what you are looking for. the surface area of the cylinder
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for )


Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The surface area is approximately 427.04 square centimeters.

Find the ⓐ volume and ⓑ surface area of a can of paint with radius 8 centimeters and height 19 centimeters. Assume the can is shaped exactly like a cylinder.

Find the ⓐ volume and ⓑ surface area of a cylindrical drum with radius 2.7 feet and height 4 feet. Assume the drum is shaped exactly like a cylinder.

Find the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.

In geometry, a<!– no-selfclose –> cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See (Figure).

Earlier in this section, we saw that the volume of a cylinder is We can think of a cone as part of a cylinder. (Figure) shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.

In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is

Since the base of a cone is a circle, we can substitute the formula of area of a circle, , for <!– no-selfclose –> to get the formula for volume of a cone.

In this book, we will only find the volume of a cone, and not its surface area.

For a cone with radius and height .

Find the volume of a cone with height inches and radius of its base inches.

Step 1. Read the problem. Draw the figure and label it
with the given information.
Step 2. Identify what you are looking for. the volume of the cone
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for )


Step 5. Solve.
Step 6. Check: We leave it to you to check your
calculations.
Step 7. Answer the question. The volume is approximately 25.12 cubic inches.

Find the volume of a cone with height inches and radius inches

Find the volume of a cone with height centimeters and radius centimeters

Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is inches tall and inches in diameter? Round the answer to the nearest hundredth.

Step 1. Read the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.
Step 2. Identify what you are looking for. the volume of the cone
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for , and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.)

Step 5. Solve.
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The volume of the wrap is approximately 52.33 cubic inches.

How many cubic inches of candy will fit in a cone-shaped piñata that is inches long and inches across its base? Round the answer to the nearest hundredth.

What is the volume of a cone-shaped party hat that is inches tall and inches across at the base? Round the answer to the nearest hundredth.

Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.

Key Concepts

  • Volume and Surface Area of a Rectangular Solid
    • For a cone with radius and height :
      Volume:

    Practice Makes Perfect

    Find Volume and Surface Area of Rectangular Solids

    In the following exercises, find ⓐ the volume and ⓑ the surface area of the rectangular solid with the given dimensions.

    length meters, width meters, height meters

    length feet, width feet, height feet

    length yards, width yards, height yards

    length centimeters, width centimeters, height centimeters

    In the following exercises, solve.

    Moving van A rectangular moving van has length feet, width feet, and height feet. Find its ⓐ volume and ⓑ surface area.

    Gift box A rectangular gift box has length inches, width inches, and height inches. Find its ⓐ volume and ⓑ surface area.

    Carton A rectangular carton has length cm, width cm, and height cm. Find its ⓐ volume and ⓑ surface area.

    Shipping container A rectangular shipping container has length feet, width feet, and height feet. Find its ⓐ volume and ⓑ surface area.

    In the following exercises, find ⓐ the volume and ⓑ the surface area of the cube with the given side length.

    centimeters

    inches

    feet

    meters

    In the following exercises, solve.

    Science center Each side of the cube at the Discovery Science Center in Santa Ana is feet long. Find its ⓐ volume and ⓑ surface area.

    Museum A cube-shaped museum has sides meters long. Find its ⓐ volume and ⓑ surface area.

    Base of statue The base of a statue is a cube with sides meters long. Find its ⓐ volume and ⓑ surface area.

    Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its ⓐ volume and ⓑ surface area.

    Find the Volume and Surface Area of Spheres

    In the following exercises, find ⓐ the volume and ⓑ the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

    centimeters

    inches

    feet

    yards

    In the following exercises, solve. Round answers to the nearest hundredth.

    Exercise ball An exercise ball has a radius of inches. Find its ⓐ volume and ⓑ surface area.

    Balloon ride The Great Park Balloon is a big orange sphere with a radius of feet . Find its ⓐ volume and ⓑ surface area.

    Golf ball A golf ball has a radius of centimeters. Find its ⓐ volume and ⓑ surface area.

    Baseball A baseball has a radius of inches. Find its ⓐ volume and ⓑ surface area.

    Find the Volume and Surface Area of a Cylinder

    In the following exercises, find ⓐ the volume and ⓑ the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.

    radius feet, height feet

    radius centimeters, height centimeters

    radius meters, height meters

    radius yards, height yards

    In the following exercises, solve. Round answers to the nearest hundredth.

    Coffee can A can of coffee has a radius of cm and a height of cm. Find its ⓐ volume and ⓑ surface area.

    Snack pack A snack pack of cookies is shaped like a cylinder with radius cm and height cm. Find its ⓐ volume and ⓑ surface area.

    Barber shop pole A cylindrical barber shop pole has a diameter of inches and height of inches. Find its ⓐ volume and ⓑ surface area.

    Architecture A cylindrical column has a diameter of feet and a height of feet. Find its ⓐ volume and ⓑ surface area.

    Find the Volume of Cones

    In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth.

    height feet and radius feet

    height inches and radius inches

    height centimeters and radius cm

    height meters and radius meters

    In the following exercises, solve. Round answers to the nearest hundredth.

    Teepee What is the volume of a cone-shaped teepee tent that is />feet tall and />feet across at the base?

    Popcorn cup What is the volume of a cone-shaped popcorn cup that is inches tall and inches across at the base?

    Silo What is the volume of a cone-shaped silo that is feet tall and feet across at the base?

    Sand pile What is the volume of a cone-shaped pile of sand that is meters tall and meters across at the base?

    Everyday Math

    Street light post The post of a street light is shaped like a truncated cone, as shown in the picture below. It is a large cone minus a smaller top cone. The large cone is feet tall with base radius foot. The smaller cone is feet tall with base radius of feet. To the nearest tenth,

    ⓐ find the volume of the large cone.

    ⓑ find the volume of the small cone.

    ⓒ find the volume of the post by subtracting the volume of the small cone from the volume of the large cone.

    Ice cream cones A regular ice cream cone is 4 inches tall and has a diameter of inches. A waffle cone is inches tall and has a diameter of inches. To the nearest hundredth,

    ⓐ find the volume of the regular ice cream cone.

    ⓑ find the volume of the waffle cone.

    ⓒ how much more ice cream fits in the waffle cone compared to the regular cone?

    Writing Exercises

    The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape.

    Which has a larger volume, a cube of sides of />feet or a sphere with a diameter of />feet? Explain your reasoning.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

    Glossary


    Prism & Formulas

    is a three dimensional plane or geometric shape having both the ends very similar, parallel and equal line lengths. A square or rectangle base with the sides of squares called the rectangle prism, whereas a triangle face with rectangle base called the triangle prism. K-12 students may refer the below formulas of pyramid to know what are all the input parameters are being used to find the volume & surface area of rectangular or triangular prism.

    Formula to calculate volume & surface area of rectangular & triangle prism


    Basic instructions for the worksheets

    Each worksheet is randomly generated and thus unique. The answer key is automatically generated and is placed on the second page of the file.

    You can generate the worksheets either in html or PDF format &mdash both are easy to print. To get the PDF worksheet, simply push the button titled "Create PDF" or "Make PDF worksheet". To get the worksheet in html format, push the button "View in browser" or "Make html worksheet". This has the advantage that you can save the worksheet directly from your browser (choose File &rarr Save) and then edit it in Word or other word processing program.

    Sometimes the generated worksheet is not exactly what you want. Just try again! To get a different worksheet using the same options:

    • PDF format: come back to this page and push the button again.
    • Html format: simply refresh the worksheet page in your browser window.

    7.3.6: Applying Volume and Surface Area

    Nelson Mathematics for Apprenticeship and Workplace
    Tables of Contents

    Nelson Mathematics for Apprenticeship and Workplace 10

    Chapter 1 – Earning Income
    Chapter Opener
    Getting Started
    1.1 Jobs and Income
    1.2 Calculating Gross Income: Salaries and Contracts
    1.3 Calculating Gross Income: Hourly Wage
    1.4 Calculating Overtime
    Mid-Chapter Review
    1.5 Calculating Gross Income: Commission
    1.6 Calculating Gross Income: Piecework
    1.7 Comparing Types of Income
    1.8 Calculating Net Pay
    1.9 Solving a Payment Puzzle
    Chapter Review
    Chapter Test

    Chapter 2 – Linear Measurement Systems
    Chapter Opener
    Getting Started
    2.1 Linear Measure in The Imperial System
    2.2 Linear Measure in The SI System
    2.3 Estimating Lengths
    Mid-Chapter Review
    2.4 Measuring Lengths
    2.5 Determining the Midpoint of a Linear Measurement
    2.6 Converting Linear Measurements:
    Imperial to SI
    2.7 Converting Linear Measurements:
    SI to Imperial
    2.8 Puzzle/Game
    Chapter Review
    Chapter Test

    Chapter 3 –Solving Linear Measurement Problems
    Chapter Opener
    Getting Started
    3.1 Working with Fractions and the Imperial System
    3.2 Using the Imperial System to Solve Problems
    Mid-Chapter Review
    3.3 Working with Decimals and the SI System
    3.4 Using the SI System to Solve Problems
    3.5 Deciding Which System to Use to Solve Problems
    3.6 Puzzle/Game
    Chapter Review
    Chapter Test

    Chapter 4 – Area Measurement
    Chapter Opener
    Getting Started
    4.1 Estimating Area Using Referents
    4.2 Estimating Area Using Grids
    4.3 Area Conversions and the Relationship Between Units
    4.4 Calculating Areas of 2–D shapes
    Mid Chapter Review
    4.5 Calculating Area of Irregular Polygons
    4.6 Calculating Area of Regular Polygons
    4.7 Calculating Area of Composite Shapes
    4.8 Calculating Surface Area of 3-D Objects
    4.9 Puzzle/Game
    Chapter Review
    Chapter Test

    Chapter 5 – Volume, Capacity, Mass and Temperature Measurement
    Chapter Opener
    Getting Started
    5.1 Volume/Capacity in The Imperial System
    5.2 Volume/Capacity in The SI System
    5.3 Mass in The Imperial System
    5.4 Mass in The SI System
    Mid-Chapter Review
    5.5 Converting Volume and Mass Measurements: Imperial to SI
    5.6 Converting Volume and Mass Measurements: SI to Imperial
    5.7 Temperature Relationships
    5.8 Puzzle/Game
    Chapter Review
    Chapter Test

    Chapter 6 – Working with Money
    Chapter Opener
    Getting Started
    6.1 Calculating Unit Price
    6.2 Calculating Sale Prices
    6.3 Determining the Best Buy
    Mid-Chapter Review
    6.4 Analyzing Sales Promotions
    6.5 Currency Exchange
    6.6 Estimating Costs in Different Countries
    6.7 Puzzle/Game
    Chapter Review
    Chapter Test

    Chapter 7 – Lines and Angles
    Chapter Opener
    Getting Started
    7.1 Estimating and Measuring Angles
    7.2 Describing Angles
    7.3 Bisecting Angles
    7.4 Replicating Angles
    Mid-Chapter Review
    7.5 Classifying Lines and Angles
    7.6 Parallel Lines and Transversals
    7.7 Calculating Angles
    7.8 Puzzle/Game
    Chapter Review
    Chapter Test

    Chapter 8 – Relationships in Right Triangles
    Chapter Opener
    Getting Started
    8.1 Understanding the Pythagorean Theorem
    8.2 Applying the Pythagorean Theorem
    8.3 Calculating Side Lengths using the Sine Ratio
    8.4 Calculating Side Lengths using the Cosine Ratio
    Mid-Chapter Review
    8.5 Calculating Side Lengths using the Tangent Ratio
    8.6 Calculating Angles
    8.7 Solving Right Triangle Problems
    8.8 Puzzle/Game
    Chapter Review
    Chapter Test

    Chapter 9 – Similar Polygons
    Chapter Opener
    Getting Started
    9.1 Similar Polygons: Angle Relationships
    9.2 Similar Polygons: Side Length Relationships
    9.3 Drawing a Similar Polygon
    Mid-Chapter Review
    9.4 Calculating Side Lengths in Similar Polygons
    9.5 Solving Problems Involving Similar Polygons
    9.6 Puzzle/Game
    Chapter Review
    Chapter Test

    Chapter 10 – Transformations
    Chapter Opener
    Getting Started
    10.1 Translations
    10.2 Rotations
    10.3 Reflections
    Mid-Chapter Review
    10.4 Dilations
    10.5 Successive Transformations
    10.6 Applications of Transformations
    10.7 Puzzle/Game
    Chapter Review
    Chapter Test

    Nelson Mathematics for Apprenticeship and Workplace 11

    Chapter 1 Interest: Investing Money
    Getting Started
    1.1 Understanding Simple Interest
    1.2 Simple Interest Problems
    1.3 Understanding Compound Interest
    Mid-Chapter Review
    1.4 Interest Game: Do They Match?
    1.5 Compounding Periods
    1.6 Compound Interest Problems
    Chapter Review
    Chapter Test

    Chapter 2 Working with Graphs
    Getting Started
    2.1 Bar Graphs
    2.2 Histograms
    2.3 Line Graphs
    2.4 Solving a Graphing Puzzle
    Mid-Chapter Review
    2.5 Circle Graphs
    2.6 Creating Graphs with Technology
    2.7 Graphic Representations
    Chapter Review
    Chapter Test

    Chapter 3 Surface Area
    Getting Started
    3.2 Surface Area: Prisms and Cylinders
    3.3 Surface Area: Pyramids and Cones
    Mid-Chapter Review
    3.4 Surface Area: Spheres
    3.5 Estimating Surface Area
    3.6 Surface Area: Dimension Changes
    3.7 Solving a Surface Area Puzzle
    Chapter Review
    Chapter Test

    Chapter 4 Volume and Capacity
    Getting Started
    4.1 Volume: Prisms, Cylinders, Pyramids, and Cones
    4.2 Volume: Spheres
    4.3 Volume: Composite Objects
    4.4 Determining Volume
    4.5 Estimating Volume
    4.6 Volume: Dimension Changes
    Mid-Chapter Review
    4.7 Solving a Three-Jug Capacity Puzzle
    4.8 Determining Capacity
    4.9 Solving Capacity Problems
    4.10 Estimating Capacity
    Chapter Review
    Chapter Test

    Chapter 5 Interest: Borrowing Money
    Getting Started
    5.1 Credit Cards
    5.2 Loans
    5.3 Lines of Credit
    Mid-Chapter Review
    5.4 Sales Promotions
    5.5 Which Do You Choose?
    Chapter Review
    Chapter Test

    Chapter 6 Slope and Rates
    Getting Started
    6.1 Slope
    6.2 Comparing Slopes
    6.3 Vertical and Horizontal Lines
    6.4 Solving Slope Problems
    6.5 Angle of Elevation and Slope
    Mid-Chapter Review
    6.6 Rates of Change
    6.7 Rates and Unit Analysis
    6.8 Conversion Within Measurement Systems
    6.9 Conversion Between Measurement Systems
    6.10 Solving a Skateboard Puzzle
    Chapter Review
    Chapter Test

    Chapter 7 Drawing Shapes and Objects
    Getting Started
    7.1 Diagrams of Objects
    7.2 Different Views of Objects
    7.3 One-Point Perspective Drawings
    7.4 Exploded View Diagrams
    Mid-Chapter Review
    7.5 Understanding Scale
    7.6 Building Scale Models
    7.7 Draw and Match Me
    7.8 Drawing Scale Diagrams
    7.9 Scale Diagrams and Technology
    Chapter Review
    Chapter Test

    Chapter 8 Managing Money
    Getting Started
    8.1 Financial Institutions
    8.2 Types of Bank Accounts
    8.3 Debit Cards
    Mid-Chapter Review
    8.4 Online Banking
    8.5 Creating a Secret Password
    8.6 Creating and Analysing a Budget
    8.7 Budgets and Technology
    Chapter Review
    Chapter Test

    Chapter 9 Solving Right Triangle Problems
    Getting Started
    9.1 Angles of Elevation
    9.2 Angles of Depression
    9.3 Solving Triangle Puzzles
    Mid-Chapter Review
    9.4 Solving Two-Triangle Problems
    9.5 Solving 3-D Problems

    Chapter Review
    Chapter Test

    Chapter 10 Linear Relations
    Getting Started
    10.1 Relations
    10.2 Graphing Linear Relations
    Mid-Chapter Review
    10.3 Solving a Dots and Lines Puzzle
    10.4 Direct and Partial Variation
    10.5 Scatter Plots
    10.6 Scatter Plots with Technology
    Chapter Review
    Chapter Test


    Glossary
    Charts and Formulas

    Nelson Mathematics for Apprenticeship and Workplace 12

    Chapter 1 Buying or Leasing a Vehicle
    Getting Started
    1.1 Buying a New Vehicle
    1.2 Buying a Used Vehicle
    1.3 Operating Costs for a Vehicle
    1.4 Who’s Buying What?
    Mid-Chapter Review
    1.5 Leasing a Vehicle
    1.6 Lease or Buy?
    1.7 Vehicle Options and Technology
    Chapter Review
    Chapter Test

    Chapter 2 – Measuring Instruments
    Getting Started
    2.1 Precision
    2.2 Precision and Calculations
    2.3 Solving a Measuring Puzzle
    Mid-Chapter Review
    2.4 Precision and Accuracy
    2.5 Uncertainty in Measurements
    Chapter Review
    Chapter Test

    Chapter 3 – Statistics
    Getting Started
    3.1 Mean
    3.2 Weighted Mean
    3.3 Median
    3.4 Mode
    3.5 Which Score is Higher?
    Mid-Chapter Review
    3.6 Interpreting Data
    3.7 Percentiles
    Chapter Review
    Chapter Test

    Chapter 4 – Linear Relations
    Getting Started
    4.1 Describing Relations
    4.2 Interpreting Linear Relations
    4.3 Direct and Partial Relations
    Mid-Chapter Review
    4.4 Equations of Linear Relations
    4.5 Creating a Number Trick
    4.6 Scatter Plots
    4.7 Scatter Plots and Technology
    Chapter Review
    Chapter Test

    Chapter 5 – Career Planning
    Getting Started
    5.1 Exploring Career Options
    5.2 Researching Your Career Choice
    5.3 Planning for Training Costs
    5.4 Writing a Resumé
    5.5 Financing Your Lifestyle
    Chapter Project

    Chapter 6 – Operating a Small Business
    Getting Started
    6.1 Business Opportunities
    6.2 Business Expenses
    6.3 Planning for Taxes
    6.4 Sidewalk Sale Game
    Mid-Chapter Review
    6.5 Improving Profitability
    6.6 Break-Even Point
    Chapter Review
    Chapter Test

    Chapter 7 – Polygons
    Getting Started
    7.1 Triangles
    7.2 Quadrilaterals
    7.3 Creating Polygon Puzzles
    Mid-Chapter Review
    7.4 Regular Polygons
    7.5 Applications of Polygons
    Chapter Review
    Chapter Test

    Chapter 8 – Transformations
    Getting Started
    8.1 Translations
    8.2 Reflections
    8.3 Rotations
    Mid-Chapter Review
    8.4 Dilations
    8.5 Dilations and Technology
    8.6 Combining 2-D Transformations
    8.7 Solving a Transformation Puzzle
    Chapter Review
    Chapter Test

    Chapter 9 – Trigonometry
    Getting Started
    9.1 Exploring the Sine Law
    9.2 Solving Sine-Law Problems
    9.3 Reversing Triangle Puzzle
    Mid-Chapter Review
    9.4 Exploring the Cosine Law
    9.5 Solving Cosine-Law Problems
    9.6 Choosing the Sine Law or Cosine Law
    Chapter Review
    Chapter Test

    Chapter 10 – Probability
    Getting Started
    10.1 Experimental Probability
    10.2 Theoretical Probability
    10.3 Three-Cup Guessing Game
    Mid-Chapter Review
    10.4 Interpreting Odds
    10.5 Making Decisions
    Chapter Review
    Chapter Test

    Chapter 11 – Owning a Home
    Getting Started
    11.1 Qualifying for a Mortgage
    11.2 Closing Costs
    11.3 Mortgage Payments
    Mid-Chapter Review
    11.4 Managing Housing Costs
    11.5 Mortgages and Technology
    11.6 Solving Map Puzzles
    Chapter Review
    Chapter Test
    Glossary


    Surface Area (BET) & Pore Size Determination (BJH)

    Surface area and pore size are of interest in many industries and processes that involve surfaces interacting with gas or liquids. Examples include sensors, catalysts, fuel cells, batteries and chemical manufacturing. The rate or volume of gas adsorption and the capacity of a material to adsorb gases can have a large effect on its functional usefulness. Investigating those factors can be extremely important during R&D, product development or later troubleshooting and failure analysis. For example, the pore size may have an effect on the rate of reaction or efficiency of a catalytic process. Similarly, the surface area of a material may have an effect on the lifetime or storage capacity of a battery, in addition to any surface chemistry effects that also occur at that surface.

    Before performing a surface area or pore size measurement, contaminants (typically water and carbon dioxide) must be removed from the solid surface. The solid is pretreated by applying heat and vacuum to remove any initially adsorbed contaminants.

    To determine the surface area, the solid is cooled, under vacuum, to cryogenic temperature (using liquid nitrogen). Nitrogen gas is dosed to the solid in controlled increments. After each dose of adsorptive gas, the pressure is allowed to equilibrate, and the quantity of gas adsorbed is determined. The quantity of gas adsorbed is plotted as a function of pressure. From this plot the quantity of gas required to form a monolayer over the external surface of the solid is determined. The surface area can be calculated from the quantity of gas required to form a monolayer, using the BET (Brunauer, Emmett and Teller) equation.

    To determine the pore volume and pore size distribution, the gas pressure is increased further incrementally until all pores are filled with liquid. Next, the gas pressure is reduced incrementally, evaporating the condensed gas from the system. Evaluation of the adsorption and desorption isotherms reveals information about the pore volume and pores size distribution. The BJH (Barrett, Joyner and Halenda) calculation is used to determine pore volume and pore size distribution.


    58 Solve Geometry Applications: Volume and Surface Area

    1. Evaluate when
      If you missed this problem, review (Figure).
    2. Evaluate when
      If you missed this problem, review (Figure).
    3. Find the area of a circle with radius
      If you missed this problem, review (Figure).

    In this section, we will finish our study of geometry applications. We find the volume and surface area of some three-dimensional figures. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.

    1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
    2. Identify what you are looking for.
    3. Name what you are looking for. Choose a variable to represent that quantity.
    4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
    5. Solve the equation using good algebra techniques.
    6. Check the answer in the problem and make sure it makes sense.
    7. Answer the question with a complete sentence.

    Find Volume and Surface Area of Rectangular Solids

    A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See (Figure)). The amount of paint needed to cover the outside of each box is the surface area , a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

    Each crate is in the shape of a rectangular solid . Its dimensions are the length, width, and height. The rectangular solid shown in (Figure) has length units, width units, and height units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.

    Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This by by rectangular solid has cubic units.

    Altogether there are cubic units. Notice that is the

    The volume, of any rectangular solid is the product of the length, width, and height.

    We could also write the formula for volume of a rectangular solid in terms of the area of the base. The area of the base, is equal to

    We can substitute for in the volume formula to get another form of the volume formula.

    We now have another version of the volume formula for rectangular solids. Let’s see how this works with the rectangular solid we started with. See (Figure).

    To find the surface area of a rectangular solid, think about finding the area of each of its faces. How many faces does the rectangular solid above have? You can see three of them.

    Notice for each of the three faces you see, there is an identical opposite face that does not show.

    The surface area of the rectangular solid shown in (Figure) is square units.

    In general, to find the surface area of a rectangular solid, remember that each face is a rectangle, so its area is the product of its length and its width (see (Figure)). Find the area of each face that you see and then multiply each area by two to account for the face on the opposite side.

    For each face of the rectangular solid facing you, there is another face on the opposite side. There are faces in all.

    For a rectangular solid with length width and height

    For a rectangular solid with length cm, height cm, and width cm, find the ⓐ volume and ⓑ surface area.

    Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    Step 2. Identify what you are looking for. the volume of the rectangular solid
    Step 3. Name. Choose a variable to represent it. Let = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.


    Step 5. Solve the equation.
    Step 6. Check
    We leave it to you to check your calculations.
    Step 7. Answer the question. The volume is cubic centimeters.
    Step 2. Identify what you are looking for. the surface area of the solid
    Step 3. Name. Choose a variable to represent it. Let = surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.


    Step 5. Solve the equation.
    Step 6. Check: Double-check with a calculator.
    Step 7. Answer the question. The surface area is 1,034 square centimeters.

    Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length feet, width feet, and height feet.

    Find the ⓐ volume and ⓑ surface area of rectangular solid with the: length feet, width feet, and height feet.

    A rectangular crate has a length of inches, width of inches, and height of inches. Find its ⓐ volume and ⓑ surface area.

    Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    Step 2. Identify what you are looking for. the volume of the crate
    Step 3. Name. Choose a variable to represent it. let = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.


    Step 5. Solve the equation.
    Step 6. Check: Double check your math.
    Step 7. Answer the question. The volume is 15,000 cubic inches.
    Step 2. Identify what you are looking for. the surface area of the crate
    Step 3. Name. Choose a variable to represent it. let = surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute.


    Step 5. Solve the equation.
    Step 6. Check: Check it yourself!
    Step 7. Answer the question. The surface area is 3,700 square inches.

    A rectangular box has length feet, width feet, and height feet. Find its ⓐ volume and ⓑ surface area.

    A rectangular suitcase has length inches, width inches, and height inches. Find its ⓐ volume and ⓑ surface area.

    Volume and Surface Area of a Cube

    A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

    So for a cube, the formulas for volume and surface area are and

    For any cube with sides of length

    A cube is inches on each side. Find its ⓐ volume and ⓑ surface area.

    Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    Step 2. Identify what you are looking for. the volume of the cube
    Step 3. Name. Choose a variable to represent it. let V = volume
    Step 4. Translate.
    Write the appropriate formula.

    Step 5. Solve. Substitute and solve.
    Step 6. Check: Check your work.
    Step 7. Answer the question. The volume is 15.625 cubic inches.
    Step 2. Identify what you are looking for. the surface area of the cube
    Step 3. Name. Choose a variable to represent it. let S = surface area
    Step 4. Translate.
    Write the appropriate formula.

    Step 5. Solve. Substitute and solve.
    Step 6. Check: The check is left to you.
    Step 7. Answer the question. The surface area is 37.5 square inches.

    For a cube with side 4.5 meters, find the ⓐ volume and ⓑ surface area of the cube.

    For a cube with side 7.3 yards, find the ⓐ volume and ⓑ surface area of the cube.

    A notepad cube measures inches on each side. Find its ⓐ volume and ⓑ surface area.

    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    Step 2. Identify what you are looking for. the volume of the cube
    Step 3. Name. Choose a variable to represent it. let V = volume
    Step 4. Translate.
    Write the appropriate formula.

    Step 5. Solve the equation.
    Step 6. Check: Check that you did the calculations
    correctly.
    Step 7. Answer the question. The volume is 8 cubic inches.
    Step 2. Identify what you are looking for. the surface area of the cube
    Step 3. Name. Choose a variable to represent it. let S = surface area
    Step 4. Translate.
    Write the appropriate formula.

    Step 5. Solve the equation.
    Step 6. Check: The check is left to you.
    Step 7. Answer the question. The surface area is 24 square inches.

    A packing box is a cube measuring feet on each side. Find its ⓐ volume and ⓑ surface area.

    A wall is made up of cube-shaped bricks. Each cube is inches on each side. Find the ⓐ volume and ⓑ surface area of each cube.

    Find the Volume and Surface Area of Spheres

    A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

    Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate with

    For a sphere with radius

    A sphere has a radius inches. Find its ⓐ volume and ⓑ surface area.

    Step 1 is the same for both ⓐ and ⓑ , so we will show it just once.

    Step 1. Read the problem. Draw the figure and label
    it with the given information.
    Step 2. Identify what you are looking for. the volume of the sphere
    Step 3. Name. Choose a variable to represent it. let V = volume
    Step 4. Translate.
    Write the appropriate formula.

    Step 5. Solve.
    Step 6. Check: Double-check your math on a calculator.
    Step 7. Answer the question. The volume is approximately 904.32 cubic inches.
    Step 2. Identify what you are looking for. the surface area of the cube
    Step 3. Name. Choose a variable to represent it. let S = surface area
    Step 4. Translate.
    Write the appropriate formula.

    Step 5. Solve.
    Step 6. Check: Double-check your math on a calculator
    Step 7. Answer the question. The surface area is approximately 452.16 square inches.

    Find the ⓐ volume and ⓑ surface area of a sphere with radius 3 centimeters.

    Find the ⓐ volume and ⓑ surface area of each sphere with a radius of foot

    A globe of Earth is in the shape of a sphere with radius centimeters. Find its ⓐ volume and ⓑ surface area. Round the answer to the nearest hundredth.

    Step 1. Read the problem. Draw a figure with the
    given information and label it.
    Step 2. Identify what you are looking for. the volume of the sphere
    Step 3. Name. Choose a variable to represent it. let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for )


    Step 5. Solve.
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question. The volume is approximately 11,488.21 cubic inches.
    Step 2. Identify what you are looking for. the surface area of the sphere
    Step 3. Name. Choose a variable to represent it. let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for )


    Step 5. Solve.
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question. The surface area is approximately 2461.76 square inches.

    A beach ball is in the shape of a sphere with radius of inches. Find its ⓐ volume and ⓑ surface area.

    A Roman statue depicts Atlas holding a globe with radius of feet. Find the ⓐ volume and ⓑ surface area of the globe.

    Find the Volume and Surface Area of a Cylinder

    If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder <!– no-selfclose –> is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height />of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, />, will be perpendicular to the bases.

    Rectangular solids and cylinders are somewhat similar because they both have two bases and a height. The formula for the volume of a rectangular solid, , can also be used to find the volume of a cylinder.

    For the rectangular solid, the area of the base, , is the area of the rectangular base, length × width. For a cylinder, the area of the base, is the area of its circular base, (Figure) compares how the formula is used for rectangular solids and cylinders.

    To understand the formula for the surface area of a cylinder, think of a can of vegetables. It has three surfaces: the top, the bottom, and the piece that forms the sides of the can. If you carefully cut the label off the side of the can and unroll it, you will see that it is a rectangle. See (Figure).

    The distance around the edge of the can is the circumference of the cylinder’s base it is also the length of the rectangular label. The height of the cylinder is the width of the rectangular label. So the area of the label can be represented as

    To find the total surface area of the cylinder, we add the areas of the two circles to the area of the rectangle.

    The surface area of a cylinder with radius and height is

    For a cylinder with radius and height

    A cylinder has height centimeters and radius centimeters. Find the ⓐ volume and ⓑ surface area.

    Step 1. Read the problem. Draw the figure and label
    it with the given information.
    Step 2. Identify what you are looking for. the volume of the cylinder
    Step 3. Name. Choose a variable to represent it. let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for )


    Step 5. Solve.
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question. The volume is approximately 141.3 cubic inches.
    Step 2. Identify what you are looking for. the surface area of the cylinder
    Step 3. Name. Choose a variable to represent it. let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for )


    Step 5. Solve.
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question. The surface area is approximately 150.72 square inches.

    Find the ⓐ volume and ⓑ surface area of the cylinder with radius 4 cm and height 7cm.

    Find the ⓐ volume and ⓑ surface area of the cylinder with given radius 2 ft and height 8 ft.

    Find the ⓐ volume and ⓑ surface area of a can of soda. The radius of the base is centimeters and the height is centimeters. Assume the can is shaped exactly like a cylinder.

    Step 1. Read the problem. Draw the figure and
    label it with the given information.
    Step 2. Identify what you are looking for. the volume of the cylinder
    Step 3. Name. Choose a variable to represent it. let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for )


    Step 5. Solve.
    Step 6. Check: We leave it to you to check.
    Step 7. Answer the question. The volume is approximately 653.12 cubic centimeters.
    Step 2. Identify what you are looking for. the surface area of the cylinder
    Step 3. Name. Choose a variable to represent it. let S = surface area
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for )


    Step 5. Solve.
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question. The surface area is approximately 427.04 square centimeters.

    Find the ⓐ volume and ⓑ surface area of a can of paint with radius 8 centimeters and height 19 centimeters. Assume the can is shaped exactly like a cylinder.

    Find the ⓐ volume and ⓑ surface area of a cylindrical drum with radius 2.7 feet and height 4 feet. Assume the drum is shaped exactly like a cylinder.

    Find the Volume of Cones

    The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.

    In geometry, a<!– no-selfclose –> cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See (Figure).

    Earlier in this section, we saw that the volume of a cylinder is We can think of a cone as part of a cylinder. (Figure) shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.

    In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is

    Since the base of a cone is a circle, we can substitute the formula of area of a circle, , for <!– no-selfclose –> to get the formula for volume of a cone.

    In this book, we will only find the volume of a cone, and not its surface area.

    For a cone with radius and height .

    Find the volume of a cone with height inches and radius of its base inches.

    Step 1. Read the problem. Draw the figure and label it
    with the given information.
    Step 2. Identify what you are looking for. the volume of the cone
    Step 3. Name. Choose a variable to represent it. let V = volume
    Step 4. Translate.
    Write the appropriate formula.
    Substitute. (Use 3.14 for )


    Step 5. Solve.
    Step 6. Check: We leave it to you to check your
    calculations.
    Step 7. Answer the question. The volume is approximately 25.12 cubic inches.

    Find the volume of a cone with height inches and radius inches

    Find the volume of a cone with height centimeters and radius centimeters

    Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is inches tall and inches in diameter? Round the answer to the nearest hundredth.

    Step 1. Read the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.
    Step 2. Identify what you are looking for. the volume of the cone
    Step 3. Name. Choose a variable to represent it. let V = volume
    Step 4. Translate. Write the appropriate formula. Substitute. (Use 3.14 for , and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.)

    Step 5. Solve.
    Step 6. Check: We leave it to you to check your calculations.
    Step 7. Answer the question. The volume of the wrap is approximately 52.33 cubic inches.

    How many cubic inches of candy will fit in a cone-shaped piñata that is inches long and inches across its base? Round the answer to the nearest hundredth.

    What is the volume of a cone-shaped party hat that is inches tall and inches across at the base? Round the answer to the nearest hundredth.

    Summary of Geometry Formulas

    The following charts summarize all of the formulas covered in this chapter.

    Key Concepts

    • Volume and Surface Area of a Rectangular Solid
      • For a cone with radius and height :
        Volume:

      Practice Makes Perfect

      Find Volume and Surface Area of Rectangular Solids

      In the following exercises, find ⓐ the volume and ⓑ the surface area of the rectangular solid with the given dimensions.

      length meters, width meters, height meters

      length feet, width feet, height feet

      length yards, width yards, height yards

      length centimeters, width centimeters, height centimeters

      In the following exercises, solve.

      Moving van A rectangular moving van has length feet, width feet, and height feet. Find its ⓐ volume and ⓑ surface area.

      Gift box A rectangular gift box has length inches, width inches, and height inches. Find its ⓐ volume and ⓑ surface area.

      Carton A rectangular carton has length cm, width cm, and height cm. Find its ⓐ volume and ⓑ surface area.

      Shipping container A rectangular shipping container has length feet, width feet, and height feet. Find its ⓐ volume and ⓑ surface area.

      In the following exercises, find ⓐ the volume and ⓑ the surface area of the cube with the given side length.

      centimeters

      inches

      feet

      meters

      In the following exercises, solve.

      Science center Each side of the cube at the Discovery Science Center in Santa Ana is feet long. Find its ⓐ volume and ⓑ surface area.

      Museum A cube-shaped museum has sides meters long. Find its ⓐ volume and ⓑ surface area.

      Base of statue The base of a statue is a cube with sides meters long. Find its ⓐ volume and ⓑ surface area.

      Tissue box A box of tissues is a cube with sides 4.5 inches long. Find its ⓐ volume and ⓑ surface area.

      Find the Volume and Surface Area of Spheres

      In the following exercises, find ⓐ the volume and ⓑ the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

      centimeters

      inches

      feet

      yards

      In the following exercises, solve. Round answers to the nearest hundredth.

      Exercise ball An exercise ball has a radius of inches. Find its ⓐ volume and ⓑ surface area.

      Balloon ride The Great Park Balloon is a big orange sphere with a radius of feet . Find its ⓐ volume and ⓑ surface area.

      Golf ball A golf ball has a radius of centimeters. Find its ⓐ volume and ⓑ surface area.

      Baseball A baseball has a radius of inches. Find its ⓐ volume and ⓑ surface area.

      Find the Volume and Surface Area of a Cylinder

      In the following exercises, find ⓐ the volume and ⓑ the surface area of the cylinder with the given radius and height. Round answers to the nearest hundredth.

      radius feet, height feet

      radius centimeters, height centimeters

      radius meters, height meters

      radius yards, height yards

      In the following exercises, solve. Round answers to the nearest hundredth.

      Coffee can A can of coffee has a radius of cm and a height of cm. Find its ⓐ volume and ⓑ surface area.

      Snack pack A snack pack of cookies is shaped like a cylinder with radius cm and height cm. Find its ⓐ volume and ⓑ surface area.

      Barber shop pole A cylindrical barber shop pole has a diameter of inches and height of inches. Find its ⓐ volume and ⓑ surface area.

      Architecture A cylindrical column has a diameter of feet and a height of feet. Find its ⓐ volume and ⓑ surface area.

      Find the Volume of Cones

      In the following exercises, find the volume of the cone with the given dimensions. Round answers to the nearest hundredth.

      height feet and radius feet

      height inches and radius inches

      height centimeters and radius cm

      height meters and radius meters

      In the following exercises, solve. Round answers to the nearest hundredth.

      Teepee What is the volume of a cone-shaped teepee tent that is />feet tall and />feet across at the base?

      Popcorn cup What is the volume of a cone-shaped popcorn cup that is inches tall and inches across at the base?

      Silo What is the volume of a cone-shaped silo that is feet tall and feet across at the base?

      Sand pile What is the volume of a cone-shaped pile of sand that is meters tall and meters across at the base?

      Everyday Math

      Street light post The post of a street light is shaped like a truncated cone, as shown in the picture below. It is a large cone minus a smaller top cone. The large cone is feet tall with base radius foot. The smaller cone is feet tall with base radius of feet. To the nearest tenth,

      ⓐ find the volume of the large cone.

      ⓑ find the volume of the small cone.

      ⓒ find the volume of the post by subtracting the volume of the small cone from the volume of the large cone.

      Ice cream cones A regular ice cream cone is 4 inches tall and has a diameter of inches. A waffle cone is inches tall and has a diameter of inches. To the nearest hundredth,

      ⓐ find the volume of the regular ice cream cone.

      ⓑ find the volume of the waffle cone.

      ⓒ how much more ice cream fits in the waffle cone compared to the regular cone?

      Writing Exercises

      The formulas for the volume of a cylinder and a cone are similar. Explain how you can remember which formula goes with which shape.

      Which has a larger volume, a cube of sides of />feet or a sphere with a diameter of />feet? Explain your reasoning.

      Self Check

      ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

      ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

      Glossary


      Body Surface Area Calculator

      The calculator below computes the total surface area of a human body, referred to as body surface area (BSA). Direct measurement of BSA is difficult, and as such many formulas have been published that estimate BSA. The calculator below provides results for some of the most popular formulas.

      Table of average BSAs

      ft 2 m 2
      Newborn child2.690.25
      Two-year-old child5.380.5
      Ten-year-old child12.271.14
      Adult female17.221.6
      Adult male20.451.9

      BSA is often used in clinical purposes over body weight because it is a more accurate indicator of metabolic mass (the body's need for energy), where metabolic mass can be estimated as fat-free mass since body fat is not metabolically active. 1 BSA is used in various clinical settings such as determining cardiac index (to relate a person's heart performance to their body size) or dosages for chemotherapy (a category of cancer treatment). While dosing for chemotherapy is often determined using a patient's BSA, there exist arguments against the use of BSA to determine medication dosages that have a narrow therapeutic index &ndash the comparison of the amount of a substance necessary to produce a therapeutic effect, to the amount that causes toxicity.

      Below are some of the most popular formulas for estimating BSA, and links to references for each for further detail on their derivations. The most widely used of these is the Du Bois formula, which has been shown to be effective for estimating body fat in both obese and non-obese patients, unlike body mass index. Where BSA is represented in m 2 , W is weight in kg, and H is height in cm, the formulas are as follows:


      Watch the video: Examples on Volume and Surface Area (December 2021).