Lesson 4: Applying Circumference

Let’s use $\pi$ to solve problems.

4.1: 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. 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 propellor with three identical propellor blades. A measuring tool starts from the center of the propellor 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?

4.2: 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
    row 1 wagon wheel   3 ft  
    row 2 airplane propeller 24 in    
    row 3 orange slice     20 cm

  2. A bug was sitting on the tip of a wind turbine blade that was 24 inches long when it 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,” and the other one that will appear, to watch the bug’s motion.

GeoGebra Applet FsKb2SXD

4.3: 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. 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.

4.4: 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. 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 horizonal 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:

$$d = 2r$$ $$C = \pi d$$ $$C = 2\pi r$$

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.

Practice Problems ▶