# 7.3.5: Distinguishing Volume and Surface Area - Mathematics

## Lesson

Let's work with surface area and volume in context.

Exercise (PageIndex{1}): THe Science Fair

Mai’s science teacher told her that when there is more ice touching the water in a glass, the ice melts faster. She wants to test this statement so she designs her science fair project to determine if crushed ice or ice cubes will melt faster in a drink.

She begins with two cups of warm water. In one cup, she puts a cube of ice. In a second cup, she puts crushed ice with the same volume as the cube. What is your hypothesis? Will the ice cube or crushed ice melt faster, or will they melt at the same rate? Explain your reasoning.

Exercise (PageIndex{2}): Revisiting the Box of Chocolates

The other day, you calculated the volume of this heart-shaped box of chocolates.

The depth of the box is 2 inches. How much cardboard is needed to create the box?

Exercise (PageIndex{3}): Card Sort: Surface Area or Volume

Your teacher will give you cards with different figures and questions on them.

1. Sort the cards into two groups based on whether it would make more sense to think about the surface area or the volume of the figure when answering the question. Pause here so your teacher can review your work.
2. Your teacher will assign you a card to examine more closely. What additional information would you need to be able to answer the question on your card?
3. Estimate reasonable measurements for the figure on your card.
4. Use your estimated measurements to calculate the answer to the question.

A cake is shaped like a square prism. The top is 20 centimeters on each side, and the cake is 10 centimeters tall. It has frosting on the sides and on the top, and a single candle on the top at the exact center of the square. You have a knife and a 20-centimeter ruler.

1. Find a way to cut the cake into 4 fair portions, so that all 4 portions have the same amount of cake and frosting.
2. Find another way to cut the cake into 4 fair portions.
3. Find a way to cut the cake into 5 fair portions.

Exercise (PageIndex{4}): A Wheelbarrow of Concrete

A wheelbarrow is being used to carry wet concrete. Here are its dimensions.

1. What volume of concrete would it take to fill the tray?
2. After dumping the wet concrete, you notice that a thin film is left on the inside of the tray. What is the area of the concrete coating the tray? (Remember, there is no top.)

### Summary

Sometimes we need to find the volume of a prism, and sometimes we need to find the surface area.

Here are some examples of quantities related to volume:

• How much water a container can hold
• How much material it took to build a solid object

Volume is measured in cubic units, like in3 or m3.

Here are some examples of quantities related to surface area:

• How much fabric is needed to cover a surface
• How much of an object needs to be painted

Surface area is measured in square units, like in2 or m2.

### 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{5})

Here is the base of a prism.

1. If the height of the prism is 5 cm, what is its surface area? What is its volume?
2. If the height of the prism is 10 cm, what is its surface area? What is its volume?
3. When the height doubled, what was the percent increase for the surface area? For the volume?

Exercise (PageIndex{6})

Select all the situations where knowing the volume of an object would be more useful than knowing its surface area.

1. Determining the amount of paint needed to paint a barn.
2. Determining the monetary value of a piece of gold jewelry.
3. Filling an aquarium with buckets of water.
4. Deciding how much wrapping paper a gift will need.
5. Packing a box with watermelons for shipping.
7. Measuring the amount of gasoline left in the tank of a tractor.

Exercise (PageIndex{7})

Han draws a triangle with a (50^{circ}) angle, a (40^{circ}) angle, and a side of length 4 cm as shown. Can you draw a different triangle with the same conditions?

(From Unit 7.2.4)

Exercise (PageIndex{8})

Angle (H) is half as large as angle (J). Angle (J) is one fourth as large as angle (K). Angle (K) has measure 240 degrees. What is the measure of angle (H)?

(From Unit 7.1.3)

Exercise (PageIndex{9})

The Colorado state flag consists of three horizontal stripes of equal height. The side lengths of the flag are in the ratio (2:3). The diameter of the gold-colored disk is equal to the height of the center stripe. What percentage of the flag is gold?

(From Unit 4.2.4)

## Step by step guide to solve Rectangular Prisms

• A Rectangular Prism is a solid (3)-dimensional object which has six rectangular faces.
• Volume of a Rectangular prism (=) Length (×) Width (×) Height
Volume (= l×w×h ) Surface area (=2(wh+lw+lh))

### Rectangular Prisms – Example 1:

Find the volume and surface area of the following rectangular prism.

Find the volume and surface area of rectangular prism.
Use volume formula: Volume (=l×w×h)
Then: Volume (=6×4×8=192) m(^3)
Use surface area formula: Surface area (=2(wh+lw+lh))
Then: Surface area (=2((4×8)+(6×4)+(6×8)))
(=2(32+24+48)=2(104)=208) m(^2)

### Rectangular Prisms – Example 2:

Find the volume and surface area of the following rectangular prism.

Find the volume and surface area of rectangular prism.
Use volume formula: Volume (=l×w×h)
Then: Volume (=10×5×8=400) m(^3)
Use surface area formula: Surface area (=2(wh+lw+lh))
Then: Surface area (=2((5×8)+(10×5)+(10×8)))
(=2(40+50+80)=340) m(^2)

## SOLVING SURFACE AREA AND VOLUME PROBLEMS

Erin is making a jewelry box of wood in the  shape of a rectangular prism. The jewelry box  will have the dimensions shown below. The cost of painting the exterior of the box is .50 per square in. How much does Erin have to spend to paint the jewelry਋ox ?

To know that total cost of painting, first we have to know the Surface area of the jewelry box.

Find surface area of the box.

Identify a base, and find its area and perimeter.

Any pair of opposite faces can be the bases. For example, we can choose the bottom and top of the box as the bases.

Find perimeter of the base.

Identify the height, and find the surface area.

The height h of the prism is 6 inches. Use the formula to find the surface area.

Total cost  =  Area x Cost per square in.

Hence, Erin has to spend $342 to paint the jewelry box. A metal box that is in the shape of rectangular prism has the following dimensions. The length is 9 inches, width is 2 inches, and height is 1 1/ 2 inches. Find the total cost of silver coating for the entire box. To know that total cost of silver coating, first we have to know the Surface area of the metal box. Find surface area of the box. Identify a base, and find its area and perimeter. Any pair of opposite faces can be the bases. For example, we can choose the bottom and top of the box as the bases. Find perimeter of the base. Identify the height, and find the surface area. The height h of the prism is 1 1/2 inches. Use the formula to find the surface area. Total cost = Area x Cost per square in. Hence, the total cost of silver coating for the entire box is$103.50.

Cherise is setting up her tent. Her tent is in the shape of a  trapezoidal prism shown below. How many cubic feet of space are in  her tent ?

To find the number of cubic feet of space in  the  tent, we have to find the volume of  Cherise's  tent.

Volume of  Cherise's  tent (Trapezoidal prism) is

Area of trapezoid with bases of lengths  b ₁ and b ₂ and height h.

Base area (b)  =  (1/2) x ( b ₁ + b ₂)h

Hence, the  number of cubic feet of space in  Cherise's  tent is 180.

Allie has two aquariums connected by  a small square prism. Find the volume  of the double aquarium.

Find the volume of each  of the larger aquariums.

Volume  =  Base area x Height

Find the volume of the connecting prism.

Volume  =  Base area x Height

Add the volumes of the three parts of the aquarium.

The volume of the aquarium is 74 cubic ft.

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## What is the Difference Between Volume and Surface Area?

Volume and surface area are two related concepts in the study of mathematics. They’re both important to understand, but equally important is understanding how they differ and what they mean. This is especially the case when it comes to computing the volume and surface areas of a prism or a cylinder.

If you think of wrapping a present in a box, you can get a good sense of how volume and surface area differ. First, you have to consider the size of the box, when you consider the size of the present. How much interior space does your box need to have so a present will fit? The measurement of the box’s capacity, how much it will hold, is its volume. Next you have to wrap the present. The amount of wrapping paper, which will cover the exterior of the box, is a very different calculation than the capacity of the box. You’ll need a separate measurement or some good guessing, to figure out the sum of the sides of all the surfaces or the surface area.

Volume of a square or rectangular box is pretty easy to compute. Simply multiply height times length times width to get the measurement. With a square it’s even easier, you merely cube one side’s length, since they all measure the same. If the side length is a, the formula is a x a x a or a 3 . When you are comparing volume and surface area, you’ll note a very different formula. You need to get the area of each face, and then add the areas of all faces together. With a square prism or cube, you’d essentially compute the area a x a or a 2 , multiplied by 6 (6a 2 ). When you’re working with a rectangular prism, you’ll have to the area the of 3 pairs of equal sides, which needed to be added together to determine surface area.

Work on volume and surface area are differ a little when you are trying to calculate the area of a cylinder. The formula for a volume of a cylinder is the area of one circular face multiplied times the height of the cylinder. It reads: πr 2 x h, or pi times the radius squared times height. Getting the surface area of the cylinder is a little trickier since the circular portion is essentially one continuous face. Computing surface area of a cylinder means computing the lateral area of this face.

Lateral area formula is the following πr2r or πd (pi times the radius doubled or pi times the diameter), multiplied to the height, πr2r x h. This is essentially the circumference of one circle times the height of the cylinder. To compute the entire formula you also need to add in the top and bottom circular faces’ areas. Since in a cylinder these are equal, the formula is 2 πr 2 . This calculation is then added to the lateral area to compute the whole surface area in the following expression:

πr2r x h + 2πr 2 = lateral area.

You can also view difference between volume and cylinder as a difference between what is inside and can be contained and the exterior of a three-dimensional object. These are valuable differences to understand in many applications, such as construction, engineering, or even present wrapping. When children complain that math is useless outside of math class, you might point out to them that knowing the difference between volume and surface area meant they got a very nicely wrapped gift for their birthday.

Tricia has a Literature degree from Sonoma State University and has been a frequent InfoBloom contributor for many years. She is especially passionate about reading and writing, although her other interests include medicine, art, film, history, politics, ethics, and religion. Tricia lives in Northern California and is currently working on her first novel.

Tricia has a Literature degree from Sonoma State University and has been a frequent InfoBloom contributor for many years. She is especially passionate about reading and writing, although her other interests include medicine, art, film, history, politics, ethics, and religion. Tricia lives in Northern California and is currently working on her first novel.

## Difference Between Volume and Surface Area

Volume and surface area are two different things in the terms of math. Math is a subject which usually deals with the figures and their geometry. There are several other branches of math which are dedicated to studies on several fronts and one of these is the geometry. In the math of geometry, the figures are observed and different calculations are made on them so that various things can be calculated about the figures. There are a host of different figures in the science of math and all of these are analysed to evaluate the properties of these various figures. The number of figures cannot be calculated as so many figures can be made by even one person. The most common figures in math are squares, rectangles, triangles, pentagons, hexagons and many others. All of the following figures which have been named above have different properties than the other and to distinguish them from each other, different parameters of the figures are calculated. Formulas in math have been devised by the scientists and the mathematicians so that they can let the people evaluate these properties of the figure and the most common of these parameters that have been made by the experts are volume and the surface area of the figure. Many people believe that the volume and the area of figures is one and the same thing but in-fact this is not the case at all. Volume and surface area of a figure are two entirely different things and there are separate procedures to evaluate these two aspects of the geometry. Volume of the figure may be described as the thing or the property of math in which the total capacity of figure of the geometry is calculated. This is a measure of how much a thing or a figure and accommodate inside it and the most common term that is used for this is the capacity of the figure. The surface area as compared to the volume of the figure is a totally different thing and it accounts for a different method to be followed.

### Instructions

The volume of a figure is the estimate of the total capacity of a figure of math. The units for this are meter cubes.

Surface Area

Surface area of a figure of math is the total area that the figure acquires on a space and this can be two dimensional and three dimensional both. The units for this are meter squares and formulas have been made to calculate this easily.

## Contents

A ball is a three-dimensional object, being the filled-in version of a sphere ("sphere" properly refers only to the surface and a sphere thus has no volume). Balls exist in any dimension and are generically called n-balls, where n is the number of dimensions.

The same reasoning can be generalized to n-balls using the general equations for volume and surface area, which are:

The surface-area-to-volume ratio has physical dimension L −1 (inverse length) and is therefore expressed in units of inverse distance. As an example, a cube with sides of length 1 cm will have a surface area of 6 cm 2 and a volume of 1 cm 3 . The surface to volume ratio for this cube is thus

For a given shape, SA:V is inversely proportional to size. A cube 2 cm on a side has a ratio of 3 cm −1 , half that of a cube 1 cm on a side. Conversely, preserving SA:V as size increases requires changing to a less compact shape.

Materials with high surface area to volume ratio (e.g. very small diameter, very porous, or otherwise not compact) react at much faster rates than monolithic materials, because more surface is available to react. An example is grain dust: while grain is not typically flammable, grain dust is explosive. Finely ground salt dissolves much more quickly than coarse salt.

A high surface area to volume ratio provides a strong "driving force" to speed up thermodynamic processes that minimize free energy.

The ratio between the surface area and volume of cells and organisms has an enormous impact on their biology, including their physiology and behavior. For example, many aquatic microorganisms have increased surface area to increase their drag in the water. This reduces their rate of sink and allows them to remain near the surface with less energy expenditure. [ citation needed ]

An increased surface area to volume ratio also means increased exposure to the environment. The finely-branched appendages of filter feeders such as krill provide a large surface area to sift the water for food. [1]

Individual organs like the lung have numerous internal branchings that increase the surface area in the case of the lung, the large surface supports gas exchange, bringing oxygen into the blood and releasing carbon dioxide from the blood. [2] [3] Similarly, the small intestine has a finely wrinkled internal surface, allowing the body to absorb nutrients efficiently. [4]

Cells can achieve a high surface area to volume ratio with an elaborately convoluted surface, like the microvilli lining the small intestine. [5]

Increased surface area can also lead to biological problems. More contact with the environment through the surface of a cell or an organ (relative to its volume) increases loss of water and dissolved substances. High surface area to volume ratios also present problems of temperature control in unfavorable environments. [ citation needed ]

The surface to volume ratios of organisms of different sizes also leads to some biological rules such as Allen's rule, Bergmann's rule [6] [7] [8] and gigantothermy. [9]

In the context of wildfires, the ratio of the surface area of a solid fuel to its volume is an important measurement. Fire spread behavior is frequently correlated to the surface-area-to-volume ratio of the fuel (e.g. leaves and branches). The higher its value, the faster a particle responds to changes in environmental conditions, such as temperature or moisture. Higher values are also correlated to shorter fuel ignition times, and hence faster fire spread rates.

A body of icy or rocky material in outer space may, if it can build and retain sufficient heat, develop a differentiated interior and alter its surface through volcanic or tectonic activity. The length of time through which a planetary body can maintain surface-altering activity depends on how well it retains heat, and this is governed by its surface area-to-volume ratio. For Vesta (r=263 km), the ratio is so high that astronomers were surprised to find that it did differentiate and have brief volcanic activity. The moon, Mercury and Mars have radii in the low thousands of kilometers all three retained heat well enough to be thoroughly differentiated although after a billion years or so they became too cool to show anything more than very localized and infrequent volcanic activity. As of April 2019, however, NASA has announced the detection of a "marsquake" measured on April 6, 2019 by NASA's InSight lander. [10] Venus and Earth (r>6,000 km) have sufficiently low surface area-to-volume ratios (roughly half that of Mars and much lower than all other known rocky bodies) so that their heat loss is minimal. [11]

## Examples for the calculation of the surface area of ​​the solid object

The lengths of the cuboid edges are in the ratio 2: 3: 4. Find their length if you know that the surface of the cuboid is 468 m 2 .
The cube has a surface of 600 cm 2 . What is its volume?
Find the edge c of cuboid if an edge a = 20 mm, b = 30 mm and surface area S = 8000 mm 2 .
The volume of the cube is 27 dm cubic. Calculate the surface of the cube.
A sphere is inscribed in the cone (the intersection of their boundaries consists of a circle and one point). The ratio of the surface of the ball and the contents of the base is 4: 3. A plane passing through the axis of a cone cuts the cone in an isoscele
The regular quadrilateral prism has a surface of 250 dm 2 , its shell has a content of 200 dm 2 . Calculate its leading edge.
Calculate the surface area of a cube in m 2 if you know that the area of its two walls is 72 dm 2 .
Find the content of the largest wall of a prism with a rectangle base with a height of 4 dm, side c = 5 cm, and side b = 6 cm.
The cylinder has a surface area of 300 square meters, while the cylinder's height is 12 m. Calculate the volume of this cylinder.
Calculate the surface of a quadrilateral prism according to the input: Area of the diamond base S1 = 2.8 m 2 , length of the base edge a = 14 dm, height of the prism 1,500 mm.
How many times does the surface of a sphere decrease if we reduce its radius twice?
Calculate the volume of the cube if its surface is 150 cm 2 .
Calculate the volume and surface of a cuboid whose edge lengths are in the ratio 2: 3: 4 and the longest edge measures 10cm.
A cuboid with dimensions of 9 cm, 6 cm, and 4 cm has the same volume as a cube. Calculate the surface of this cube.
Find the surface and volume of a cuboid whose dimensions are 1 m, 50 cm, and 6 dm.
The cube has a surface area of 216 dm 2 . Calculate: a) the content of one wall, b) edge length, c) cube volume.
The surface of the regular quadrilateral prism is 8800 cm 2 , the base edge is 20 cm long. Calculate the volume of the prism
In a regular quadrilateral pyramid, the height of the sidewall is equal to the length of the edge of the base. The content of the sidewall is 32 cm 2 . What is the surface of the pyramid?
Eight small Christmas balls with a radius of 1 cm have the same volume as one large Christmas ball. What has a bigger surface: eight small balls, or one big ball?
Calculate how much cover (without a floor) is used to make a tent that has the shape of a regular square pyramid. The edge of the base is 3 m long and the height of the tent is 2 m.

## Example

Each student will be given 10 cubes, though they will be working through the problem set with partners. Students are expected to solve the problems in the manner of the example: 1) build the model 2) draw the 2D views 3) find the surface area and volume 4) Examine the relationship between the 2D views and the surface area. While students are working, I will walk around to help students who are struggling with the models. I will also be looking for proper labeling of units (MP6) and accurate 2D views. It is important to let the students know that there are no "holes" or missing cubes hidden from view. For example, in problem a there are only 3 visible cubes, but the top cube is sitting on stop of a hidden cube. It can be tricky to keep track of the faces when counting for surface area. I may let students who are getting frustrated to lightly mark each face using a wet erase marker.

Page 2 of the model, labeled "independent problem solving", has the extension problems. These will present a bit more of a challenge for students who make quick work out of the first 6 problems. I will give these students another 10 cubes to work with.

## Practical Problems

A wall of length 10 m was to be built across an open ground. The height of the wall is 4 m and thickness of the wall is 24 cm. If this wall is to be built up with bricks whose dimensions are 24 cm × 12 cm × 8 cm, how many bricks would be required ?

Since the wall with all its bricks makes up the space occupied by it, we need to find the volume of the wall, which is nothing but a cuboid.

Here, Length = 10 m = 1000 cm

Volume of the wall = length × thickness × height

Now, each brick is a cuboid with length = 24 cm, breadth = 12 cm and height = 8 cm

So, volume of each brick = length × breadth × height

So, number of bricks required =

volume of the wall/volume of each brick

=  (1000 x 24 x 400) / (24 x 12 x 8)

So, the wall requires 4167 bricks.

At a Ramzan Mela, a stall keeper in one  of the food stalls has a large cylindrical vessel of base  radius 15 cm filled up to a height of 32 cm with orange  juice. The juice is filled in small cylindrical glasses  of radius 3 cm up to a height of 8 cm . Find the number of glasses that he can fill the  juice completely?

The volume of juice in the vessel  = volume of the cylindrical vessel   =    πr 2 h

Similarly, the volume of juice each glass can hold =    π r ²h

(where r and h are taken as the radius and height respectively of each glass)

So, number of glasses of juice that are sold

=   volume of the vessel/ volume of each glass

=  ( π  x 15 x 15 x 32) / ( π x ਃ x 3 x 8)

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## Video – Lesson & Examples

• Introduction to prisms
• 00:00:23 – What is a prism? formulas for lateral area, surface area, and volume
• Exclusive Content for Member’s Only
• 00:10:38 – Find the surface area and volume for each prism (Examples #1-6)
• 00:17:16 – Use the volume addition postulate to find the volume of the polyhedron (Examples #7-8)
• 00:17:16 – Determine the number of gallons of water in a swimming pool given its volume (Example #9)
• 00:17:16 – Find the surface area of a right rectangular prism (Example #10)
• 00:17:16 – Find the surface area of an oblique prism (Example #11)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions