# 3: Measuring Circles - Mathematics

3: Measuring Circles - Mathematics

## Calculate All Circle Measurements (A)

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The middle stanza of Soddy's Kiss Precise gives the formula:

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

Applied here it says $frac 3+frac 1=frac 12 left(frac 3r-frac 1R ight)^2frac 3+frac 1=frac <9><2r^2>+frac 1<2R^2>-frac 3$ As all the we can get is the ratio, let $r=1$ and we have $3+frac 1=frac 92+frac 1<2R^2>-frac 3R=3R^2-6R-1R=frac 16(6pmsqrt<48>)=frac 13(3pm2sqrt<3>)$ and we want the plus sign. Call the radius of the smaller circles $r$. Their centers form an equilateral triangle of side $2 r$. The centers of the small circles are at a distance of $R - r$ from the large circle's center, and $r$ from the large circumference. The tangency points of the smaller and large circle are also an equilateral triangle. I believe that drawing all the triangles mentioned gives you enough in terms of angles to find relations among $r$, $R$ and $R - r$ to get $r$ by trigonometry.

Note: There is also a circle internally tangent to the three tangent circles.

(I get (-3 + 2 √3)R = r for the relationship between the outer circle of radius R to the inner circle(s) of radius r.) Two more instances of same question:

The inner/outer circle is $3 pm 2 sqrt<3>$ times the radius of the three circles.

For a ring of n circles inscribed within a larger circle this will calculate the radius or diameter for either the small circles or the larger circle.

$n$ = the number of small circles
$r$ = the radius of the small circles
$R$ = the Radius of the large perimeter circle formed by the outer ring of small circles
$d$ = the diameter of the small circles
$D$ = the Diameter of the large perimeter circle formed by the outer ring of small circles

$r = R ⋅ sin(π ÷ n) ÷ [sin(π ÷ n) + 1]$
$R = r ⋅[sin(π ÷ n) + 1] ÷ sin(π ÷ n)$

$d = D ⋅ sin(π ÷ n) ÷ [sin(π ÷ n) + 1]$
$D = d ⋅[sin(π ÷ n) + 1] ÷ sin(π ÷ n)$

Note: the Radians function within Excel is not used for these formulas.

I strongly recommend drawing out the diagram I am going to explain, with the initial diagram given in mind, as even I wouldn't be able to imagine this from my explanation. With that being said,

The circle is a plane shape (two dimensional), so:

Circle: the set of all points on a plane that are a fixed distance from a center.

The area of a circle is &pi times the radius squared, which is written:

To help you remember think "Pie Are Squared" (even though pies are usually round) :

### Area Compared to a Square

A circle has about 80% of the area of a similar-width square.
The actual value is ( &pi /4) = 0.785398. = 78.5398. %

And something interesting for you:

## Circles

These are ovals. They are
symmetric and round, but they
are still not circles. Why not?

In a circle, the distance from the center point to the actual
circle line, or circumference of the circle, remains the same.

This distance is called the radius of the circle.

1. Draw a radius or a diameter from the given point. Use a ruler. Look at the example.

2. Learn to use a compass to draw circles.

a. Draw many circles with the compass.

b. Now, set the radius on the compass to be 3 cm, and draw a circle.
You can do that by placing the compass next to a ruler, and adjusting
the radius of the compass until it is 3 cm as measured by the ruler.
Some compasses show the radius for you, so you won't need a ruler.

c. Draw a circle with a radius of 5 cm.

d. Draw a circle with a radius of 1 ½ in.

3. a. Draw two diagonals into this square. Draw a point
where they cross (the center point of the square).
Now, erase the lines you drew, leaving the point.

b. Draw a circle around the square so that it touches
the vertices of the square. Use the point you drew
in (a) as the center point.

4. a. Draw a circle inside this square so that it touches
the sides of the square but will not cross over them.

b. Fill in: The _____________________ of the square
has the same length as the diameter of the circle.

5. a. Draw a circle with center point (5, 6)
and a radius of 2 units. Use a compass.

b. Draw another circle with the same center

6. Draw these figures using a compass and a ruler only in your notebook. The copies you draw do
not have to be the same exact size as here they just need to show the same pattern. See hints at
the bottom of this page. Optionally, you can also draw these in drawing software.

a. Hint: Draw a line. Then, draw the three center points on it, equally spaced.

b. Hint: First, draw the three center points for the three circles, equally spaced.
What is the radius of the big circle compared to the radius of the small ones?

c. Hint: What pattern is there in the radii of these circles? These circles are called concentric circles because they share the same center point.

d. Hint: You need to draw the outer square first. Then measure and divide it into quarters. Measure
to draw the center points of the circles (they are midpoints of the sides of the smaller squares).

This lesson is taken from Maria Miller's book Math Mammoth Geometry 1, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller. #### Math Mammoth Geometry 1

A self-teaching worktext for 4th-5th grade that covers angles, triangles, quadrilaterals, cirlce, symmetry, perimeter, area, and volume. Lots of drawing exercises!

## How to Solve a Circle Problem

Now that you know your formulas, let’s walk through the SAT math tips and strategies for solving any circle problem that comes your way.

#1: Remember your formulas and/or know where to look for them

As we mentioned earlier, it is always best to remember your formulas when you can. But if you don’t feel comfortable memorizing formulas or you fear you will mix them up, don’t hesitate to look to your formula box--that is exactly why it is there.

Just be sure to look over the formula box before test day so that you know exactly what is on it, where to find it, and how you can use that information. (For more on the formulas you are given on the test, check out our guide to SAT math formulas .)

#2: Draw, draw, draw

If you’re not given a diagram, draw one yourself! It doesn’t take long to make your own picture and doing so can save you a lot of grief and struggle as you go through your test. It can be all too easy to make an assumption or mix up your numbers when you try to perform math in your head, so don’t be afraid to take a moment to draw your own pictures.

And when you are given a diagram, draw on it too! Mark down congruent lines and angles, write in your radius measurement or your given angles. Mark any and all pieces of information you need or are given. The reason not everything is marked in your diagrams is so that the question won’t be too easy, so always write in your information yourself.

#3: Analyze what’s really being asked of you

All the formulas in the world won’t help you if you think you’re supposed to find the area, but you’re really being asked to find the circumference. Always remember that standardized tests are trying to get you to solve questions in ways in which you’re likely unfamiliar, so read carefully and pay close attention to the question you’re actually being asked.

Once you’ve verified what you’re supposed to find, most circle questions are fairly straightforward. Plug your givens into your formulas, isolate your missing information, and solve. Voila!

## 3.2: Measuring Circumference and Diameter (25 minutes)

### Activity

In this activity, students measure the diameter and circumference of different circular objects and plot the data on a coordinate plane, recalling the structure of the first activity in this unit where they measured different parts of squares. Students use a graph in order to conjecture an important relationship between the circumference of a circle and its diameter (MP 5). They notice that the two quantities appear to be proportional to each other. Based on the graph, they estimate that the constant of proportionality is close to 3 (a table of values shows that it is a little bigger than 3). This is their first estimate of pi.

This activity provides good, grade-appropriate evidence that there is a constant of proportionality between the circumference of a circle and its diameter. Students will investigate this relationship further in high school, using polygons inscribed in a circle for example.

To measure the circumference, students can use a flexible measuring tape or a piece of string wrapped around the object and then measure with a ruler. As students measure, encourage them to be as precise as possible. Even so, the best precision we can expect for the proportionality constant in this activity is “around 3” or possibly “a little bit bigger than 3.” This could be a good opportunity to talk about how many digits in the answer is reasonable. To get a good spread of points on the graph, it is important to use circles with a wide variety of diameters, from 3 cm to 25 cm. If there are points that deviate noticeably from the overall pattern, discuss how measurement error plays a factor.

As students work, monitor and select students who notice that the relationship between diameter and circumference appears to be proportional, and ask them to share during the whole-group discussion.

If students are using the digital version of the activity, they don’t necessarily need to measure physical objects, but we recommend they do so anyway.

### Launch

Arrange students in groups of 2–4. Distribute 3 circular objects and measuring tapes or string and rulers to each group. Especially if using string and rulers, it may be necessary to demonstrate the method for measuring the circumference.

Ask students to complete the first two questions in their group, and then gather additional information from two other groups (who measured different objects) for the third question.

If using the digital activity, students can work in groups of 2–4. They only need the applet to generate data for their investigation.

Your teacher will give you several circular objects.

Explore the applet to find the diameter and the circumference of three circular objects to the nearest tenth of a unit. Record your measurements in the table.

Plot the diameter and circumference values from the table on the coordinate plane. What do you notice?

### Launch

Arrange students in groups of 2–4. Distribute 3 circular objects and measuring tapes or string and rulers to each group. Especially if using string and rulers, it may be necessary to demonstrate the method for measuring the circumference.

Ask students to complete the first two questions in their group, and then gather additional information from two other groups (who measured different objects) for the third question.

If using the digital activity, students can work in groups of 2–4. They only need the applet to generate data for their investigation.

Your teacher will give you several circular objects.

Measure the diameter and the circumference of the circle in each object to the nearest tenth of a centimeter. Record your measurements in the table.

Plot the diameter and circumference values from the table on the coordinate plane. What do you notice?  Expand Image

Description: <p>A coordinate plane with the origin labeled "O". The horizontal axis is labeled "diameter, in centimeters," and the numbers 0 through 25, in increments of 5, are indicated. The vertical axis is labeled "circumference, in centimeters," and the numbers 0 through 80, in increments of 10, are indicated.</p> Plot the points from two other groups on the same coordinate plane. Do you see the same pattern that you noticed earlier?

### Anticipated Misconceptions

Students may try to measure the diameter without going across the widest part of the circle, or may struggle with measuring around the circumference. Mentally check that their measurements divide to get approximately 3 or compare with your own prepared table of data and prompt them to re-measure when their measurements are off by too much. If the circular object has a rim or lip, this could help students keep the measuring tape in place while measuring the circumference.

If students are struggling to see the proportional relationship, remind them of recent examples where they have seen similar graphs of proportional relationships. Ask them to estimate additional diameter-circumference pairs that would fit the pattern shown in the graph. Based on their graphs, do the values of the circumferences seem to relate to those of the diameters in a particular way? What seems to be that relationship?

### Activity Synthesis

Display a graph for all to see, and plot some of the students’ measurements for diameter and circumference. In cases where the same object was measured by multiple groups, include only one measurement per object. Ask students to share what they notice and what they wonder about the graph.

• Students may notice that the measurements appear to lie on a line (or are close to lying on a line) that goes through ((0, 0)) . If students do not mention a proportional relationship, make this explicit.
• Students may wonder why some points are not on the line or what the constant of proportionality is.

Invite students to estimate the constant of proportionality. From the graph, it may be difficult to make a better estimate than about 3. Another strategy is to add a column to the table, and compute the quotient of the circumference divided by the diameter for each row. For example,

object diameter (cm) circumference (cm) ( ext div ext)
soup can 6.8 21.5 3.2
tomato paste can 5.4 17 3.1
tuna can 8.5 26.5 3.1

Ask students why these numbers might not be exactly the same (measurement error, rounding). Use the average of the quotients, rounded to one or two decimal places, to come up with a “working value” of the constant of proportionality: for the numbers in the sample table above, 3.1 would be an appropriate value. This class-generated proportionality constant will be used in the next activity, to help students understand how to compute circumference from diameter and vice versa. There’s no need to mention pi or its usual approximations yet.

Time permitting, it could be worth discussing accuracy of measurements for circumference and diameter. Measuring the diameter to the nearest tenth of a centimeter can be done pretty reliably with a ruler. Measuring the circumference of a circle to the nearest tenth of a centimeter may or may not be reliable, depending on the method used. Wrapping a flexible measuring tape around the object is likely the most accurate method for measuring the circumference of a circle.

## 3 Act Math

The 3 Act Math format was developed by Dan Meyer. See the links below.

Water Leak Will service arrive in time? Car Ramp How far did the car travel? Apple Park How long to walk around? Closest to the Pin Which ball is closer?
Night Eyes How many eyes?
100,000 How many days?
Wheel of Fortune How many seconds?
3 Shapes How many points?
Cups How many cups?
Nardo Ring Which car will win?
Ping Pong Lottery Which # was pulled?
Air Mattress How long to air up?
Mr. Clean Is it really 20% more?
Record Flip Will he make the jump?
Water Filter How long will it last?
Pancakes How many pancakes?
Donut Holes How many donut holes?
Paul Sturgess What fractions?
Gym Choice Which gym is best? How many will fit?
Life-Size Jenga How many boards?
Puncher’s Chance Will he break record?
King Clutch Who is most clutch?
Gas Pump How much money? Dominoes How long will it take?
Timer How much time is left? Captain’s Wheel How many degrees? Equidistant Arena Where will it be?
Virginia Museum How many parking spots?
Brakes! How many mph?
Pac-Man Which route?
Sine WaveRunner Who will win?
Fore Right! How long is the shot?
Central Park What % of Manhattan?
Royal Flush Will he get the flush?
Kerbey Lane How many pancakes?
One-Rep Max What is his 1-rep max? Will it fit in the bag?
Energy Drinks How many to be equal?
Baby Room How many paint cans?
Elevator or Stairs? Which way is faster?
Crane Rescue How many stories?
26’s Will the rims fit?
Pop Top How many bags will fit?
Sam Houston How many real Sams?
Record Breaker Will they score enough?
Boy or Girl? What will the baby be? Is there enough gas? Moon Rise How long will it take?
Calculator Countdown How many times?
Lava Field How long will it take?
Water Tower How tall is the light?
Game Winner Who won?
Putt Putt Will it go in?
Circles Which will fill first? Will it download in time?
52 Card Pickup? How many cards?
Crescent Dunes How many mirrors?
Kerbey Lane (Part 2) How many steps?
Three Bridges Which car will win?
98 Pizzas Will the pizzas fit?
Waffles Which is a better value?
Rotonda West How many houses?
Happy Meals How many will fit?
Beats to West Which beat is fastest?
Brick by Brick How many bricks?
Snack Packs How many pretzels?
Nail Polish Will she have time?
Center Stage Will the court fit?
Commercial Break Does he have time?
M&M’s How many calories?
Jack in the Box Who will get there first?

A circle is all points in the same plane that lie at an equal distance from a center point. The circle is only composed of the points on the border. You could think of a circle as a hula hoop. It's only the points on the border that are the circle. The points within the hula hoop are not part of the circle and are called interior points.

The distance between the midpoint and the circle border is called the radius. A line segment that has the endpoints on the circle and passes through the midpoint is called the diameter. The diameter is twice the size of the radius. A line segment that has its endpoints on the circular border but does not pass through the midpoint is called a chord. The distance around the circle is called the circumference, C, and could be determined either by using the radius, r, or the diameter, d:

A circle is the same as 360°. You can divide a circle into smaller portions. A part of a circle is called an arc and an arc is named according to its angle. Arcs are divided into minor arcs (0° < v < 180°), major arcs (180° < v < 360°) and semicircles (v = 180°).

The length of an arc, l, is determined by plugging the degree measure of the Arc, v, and the circumference of the whole circle, C, into the following formula:

When diameters intersect at the central of the circle they form central angles. Like when you cut a cake you begin your pieces in the middle. As in the cake above we divide our circle into 8 pieces with the same angle. The circumference of the circle is 20 length units. Determine the length of the arc of each piece.

First we need to find the angle for each piece, since we know that a full circle is 360° we can easily tell that each piece has an angle of 360/8=45°. We plug these values into our formula for the length of arcs:

Hence the length of our arcs are 2.5 length units. We could even easier have told this by simply diving the circumference by the number of same size pieces: 20/8=2.5

## 3: Measuring Circles - Mathematics

a)measure the distance around a polygon in order to determine perimeter and
b)count the number of square units needed to cover a given surface in order to determine area.

Computation and Estimation

Probability, Statistics, Patterns, Functions & Algebra

Compare two objects/events/with nonstandard units-length/height/weight

a.) compare volumes of two containers

b.) weight/mass of two objects

Polygon: a closed plane figure composed of line segments that do not cross.

Perimeter:
a measure of the distance around a polygon and is found by adding the measures of the sides.

Area: the number of square units needed to cover a surface.

Spaghetti and Meatballs for all!

Curriculum Framework 2009

Understanding the Standard

Essential Understandings

Essential Knowledge and Skills

• A polygon is a closed plane figure composed of at least three line segments that do not cross.None of the sides are curved.

· Perimeter is a measure of the distance around a polygon and is found by adding the measures of the sides.

· Area is the number of iterations of a two-dimensional unit needed to cover a surface.The two-dimensional unit is usually a square, but it could also be another shape such as a rectangle or an equilateral triangle.

• Opportunities to explore the concepts of perimeter and area should involve hands-on experiences (e.g., placing tiles (units) around a polygon and counting the number of tiles to determine its perimeter and filling or covering a polygon with cubes (square units) and counting the cubes to determine its area).

· Understand the meaning of a polygon as a closed figure with at least three sides.None of the sides are curved and there are no intersecting lines.

· Understand that perimeter is a measure of the distance around a polygon.

· Understand how to determine the perimeter by counting the number of units around a polygon.

· Understand that area is a measure of square units needed to cover a surface.

· Understand how to determine the area by counting the number of square units.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

· Measure each side of a variety of polygons and add the measures of the sides to determine the perimeter of each polygon.

· Determine the area of a given surface by estimating and then counting the number of square units needed to cover the surface.