3.1.4: Applying Circumference

Lesson

Let's use \(\pi\) to solve problems.

Exercise \(\PageIndex{1}\): What Do We Know? What Can we Estimate?

Here are some pictures of circular objects, with measurement tools shown. The measurement tool on each picture reads as follows:

• Wagon wheel: 3 feet
• Plane propeller: 24 inches
• Sliced Orange: 20 centimeters

Figure \(\PageIndex{1}\): A picture of three different circular objects. The leftmost object is a wagon wheel with a measuring tool starting from one point on the wheel, goes through the wheel center to a point on the other side of the wheel. The center object is a plane propeller with three identical propeller blades. A measuring tool starts from the center of the propeller and goes to the end of the blade. The third object is of a sliced orange. A measuring tool goes around the entire circular region of the orange.

1. For each picture, which measurement is shown?
2. Based on this information, what measurement(s) could you estimate for each picture?

Exercise \(\PageIndex{2}\): Using \(\pi\)

In the previous activity, we looked at pictures of circular objects. One measurement for each object is listed in the table.

Your teacher will assign an approximation for \(\pi\) for you to use in this activity.

1. Complete the table.

   Table \(\PageIndex{1}\)

   object	radius	diameter	circumference
   wagon wheel		\(3\) ft
   airplane propeller	\(24\) in
   orange slice			\(20\) cm

2. A bug was sitting on the tip of the propeller blade when the propeller started to rotate. The bug held on for 5 rotations before flying away. How far did the bug travel before it flew off?

• If you choose to, you can change the settings in the applet and enter your calculation in the box at the bottom to check your work.
• Just for fun, use the slider marked "turn" to watch the bug's motion.

Exercise \(\PageIndex{3}\): Around the Running Track

The field inside a running track is made up of a rectangle that is 84.39 m long and 73 m wide, together with a half-circle at each end.

Figure \(\PageIndex{2}\): A picture of a field inside a running track. The field inside the track is composed of a rectangle, indicated by two dashed vertical lines labeled 73 meters and a horizontal length labeled 84 point 3 9 meters. There is a semi circle on each vertical side of the rectangle. The running track goes completely around the field and has a width of 9 point 7 6 meters.

1. What is the distance around the inside of the track? Explain or show your reasoning.
2. The track is 9.76 m wide all the way around. What is the distance around the outside of the track? Explain or show your reasoning.

Are you ready for more?

This size running track is usually called a 400-meter track. However, if a person ran as close to the “inside” as possible on the track, they would run less than 400 meters in one lap. How far away from the inside border would someone have to run to make one lap equal exactly 400 meters?

Exercise \(\PageIndex{4}\): Measuring a Picture Frame

Kiran bent some wire around a rectangle to make a picture frame. The rectangle is 8 inches by 10 inches.

Figure \(\PageIndex{3}\): A rectangular figure that represents a picture frame. On each vertex is a three-quarters-circle where each vertex is the center of the three-quarters-circle. There are 3 identical half-circles on each horizontal length and 4 identical half-circles on each vertical width.

1. Find the perimeter of the wire picture frame. Explain or show your reasoning.
2. If the wire picture frame were stretched out to make one complete circle, what would its radius be?

Summary

The circumference of a circle, \(C\), is \(\pi\) times the diameter, \(d\). The diameter is twice the radius, \(r\). So if we know any one of these measurements for a particular circle, we can find the others. We can write the relationships between these different measures using equations:

\(d=2r\qquad C=\pi d\qquad C=2\pi r\)

If the diameter of a car tire is 60 cm, that means the radius is 30 cm and the circumference is \(60\cdot\pi\) or about 188 cm.

If the radius of a clock is 5 in, that means the diameter is 10 in, and the circumference is \(10\cdot\pi\) or about 31 in.

If a ring has a circumference of 44 mm, that means the diameter is \(44\div\pi\), which is about 14 mm, and the radius is about 7 mm.

Glossary Entries

Definition: Circle

A circle is made out of all the points that are the same distance from a given point.

For example, every point on this circle is 5 cm away from point \(A\), which is the center of the circle.

Figure \(\PageIndex{4}\)

Definition: Circumference

The circumference of a circle is the distance around the circle. If you imagine the circle as a piece of string, it is the length of the string. If the circle has radius \(r\) then the circumference is \(2\pi r\).

The circumference of a circle of radius 3 is \(2\cdot\pi\cdot 3\), which is \(6\pi\), or about 18.85.

Definition: Diameter

A diameter is a line segment that goes from one edge of a circle to the other and passes through the center. A diameter can go in any direction. Every diameter of the circle is the same length. We also use the word diameter to mean the length of this segment.

Figure \(\PageIndex{5}\)

Definition: pi (\(\pi\))

There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is \(\pi\).

We can represent this relationship with the equation \(C=\pi d\), where \(C\) represents the circumference and \(d\) represents the diameter.

Some approximations for \(\pi\) are \(\frac{22}{7}\), 3.14, and 3.14159.

Figure \(\PageIndex{6}\)

Definition: Radius

A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.

For example, \(r\) is the radius of this circle with center \(O\).

Figure \(\PageIndex{7}\)

Practice

Exercise \(\PageIndex{5}\)

Here is a picture of a Ferris wheel. It has a diameter of 80 meters.

Figure \(\PageIndex{8}\): "Steiger Ferris Wheel 1102009 1”, by Zonk43. Public Domain. Wikimedia Commons. Source.

1. On the picture, draw and label a diameter.
2. How far does a rider travel in one complete rotation around the Ferris wheel?

Exercise \(\PageIndex{6}\)

Identify each measurement as the diameter, radius, or circumference of the circular object. Then, estimate the other two measurements for the circle.

1. The length of the minute hand on a clock is 5 in.
2. The distance across a sink drain is 3.8 cm.
3. The tires on a mining truck are 14 ft tall.
4. The fence around a circular pool is 75 ft long.
5. The distance from the tip of a slice of pizza to the crust is 7 in.
6. Breaking a cookie in half creates a straight side 10 cm long.
7. The length of the metal rim around a glass lens is 190 mm.
8. From the center to the edge of a DVD measures 60 mm.

Exercise \(\PageIndex{7}\)

A half circle is joined to an equilateral triangle with side lengths of 12 units. What is the perimeter of the resulting shape?

Figure \(\PageIndex{9}\): The figure shows a half circle and a triangle. The triangle is drawn adjacent and to the right of the half circle such that the dashed vertical base of the triangle is the diameter of the half circle.

Exercise \(\PageIndex{8}\)

Circle A has a diameter of 1 foot. Circle B has a circumference of 1 meter. Which circle is bigger? Explain your reasoning. (1 inch = 2.54 centimeters)

Exercise \(\PageIndex{9}\)

The circumference of Tyler's bike tire is 72 inches. What is the diameter of the tire?

(From Unit 3.1.3)