Lesson 13

Applying Circumference

Let’s use \(\pi\) to solve problems.

13.1: 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
A picture of three different circular objects. 

  1. For each picture, which measurement is shown?

  2. Based on this information, what measurement(s) could you estimate for each picture?

13.2: 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.
    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.

13.3: 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.

A picture of a field inside a running track.
  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.


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?

13.4: Measuring a Picture Frame

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

A rectangular figure that represents a picture frame.
  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:

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

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

If the radius of a clock is 5 in, that means the diameter is 10 in, and the circumference is \(10 \boldcdot \pi\) 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

  • 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.

    A circle with points A, B, C, D, E, F
  • 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 \(2 \boldcdot \pi \boldcdot 3\), which is \(6\pi\), or about 18.85. 

  • 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.

    A circle with its diameter labeled
  • 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.

    a graph in the coordinate plane
  • 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 \(O\).

    a circle with a labeled radius