Lesson 3: Exploring Circumference

Let’s explore the circumference of circles.

3.1: Which Is Greater?

Clare wonders if the height of the toilet paper tube or the distance around the tube is greater. What information would she need in order to solve the problem? How could she find this out?

Three different views of a circular shaped toilet paper tube. The first view is of the vertical height of the tube. The second view is of the circular base of the tube. The third view is of both the base and the height of the tube.

3.2: Measuring Circumference and Diameter

Coins, cookies, and drinking glasses are some examples of common circular objects.

  1. 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. 

    GeoGebra Applet FRh5M5eb

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

    GeoGebra Applet ju3eMb9f

  1. Plot the points from two other groups on the same coordinate plane. Do you see the same pattern that you noticed earlier?

3.3: Calculating Circumference and Diameter

Here are five circles. One measurement for each circle is given in the table.

Five circles, each with a different diameter, are labeled A, B, C, D, and E.

Use the constant of proportionality estimated in the previous activity to complete the table.

  diameter (cm) circumference (cm)
circle A 3  
circle B 10  
circle C   24
circle D   18
circle E 1  


There is a proportional relationship between the diameter and circumference of any circle. That means that if we write $C$ for circumference and $d$ for diameter, we know that $C=kd$, where $k$ is the constant of proportionality.

The exact value for the constant of proportionality is called $\boldsymbol\pi$. Some frequently used approximations for $\pi$ are $\frac{22} 7$, 3.14, and 3.14159, but none of these is exactly $\pi$.

A graph of a line in the coordinate plane with the origin labeled O. The horizontal axis is labeled “d” and the numbers 1 through 6 are indicated. The vertical axis is labeled “C” and the numbers 2 through 12, in increments of 2, are indicated. The line begins at the origin, slants upward and to the right, and passes through the point 1 comma pi.

We can use this to estimate the circumference if we know the diameter, and vice versa. For example, using 3.1 as an approximation for $\pi$, if a circle has a diameter of 4 cm, then the circumference is about $(3.1)\boldcdot 4 = 12.4$ or 12.4 cm.

The relationship between the circumference and the diameter can be written as

$$C = \pi d$$

Practice Problems ▶


pi ($\pi$)

pi ($\pi$)

The Greek letter $\pi$ (pronounced "pie") stands for the number that is the constant of proportionality between the circumference of a circle and its diameter. If $d$ is the diameter and $C$ is the circumference, then $C = \pi d$.