Math Core

A bike wheel, a round table, a pizza: circles are everywhere, and two measurements come up again and again. The circumference is the distance around a circle, and the area is the space inside it. Both come from one length, the radius, and both use the number $\pi$ .

Parts of a circle

Definition

Radius, diameter and circumference

The radius $r$ is the distance from the center to any point on the circle. The diameter $d$ is the distance straight across through the center, so $d = 2r$ . The circumference $C$ is the distance around the circle.

A circle with radius 5 cm. Its diameter is 10 cm.

The number π

Measure the circumference and diameter of any circle, a coin or a hula hoop, and divide. You always get the same number, a little more than $3$ . That number is called $\pi$ (pi):

\pi = \frac{C}{d} \approx 3.14159\ldots

Its digits go on forever without repeating, so we use $3.14$ or $\dfrac{22}{7}$ as approximations. When a problem says "in terms of $\pi$ ," leave the $\pi$ in your answer, like $10\pi$ . That answer is exact.

Circle formulas

C = \pi d = 2\pi r \qquad\qquad A = \pi r^2

Circumference is a length (units). Area is measured in square units.

Worked example: Both measurements from the radius

For the circle above, with $r = 5$ cm, find the circumference and area in terms of $\pi$ , then approximate each using $3.14$ .

\begin{aligned} C &= 2\pi r = 2\pi(5) = 10\pi \approx 31.4 \text{ cm} \\ A &= \pi r^2 = \pi(5^2) = 25\pi \approx 78.5 \text{ cm}^2 \end{aligned}

Common mistake

In $\pi r^2$ , only the radius is squared. Square $r$ first, then multiply by $\pi$ . And if you're given the diameter, cut it in half before using $A = \pi r^2$ .

Working backward

In Pre-Algebra you can treat a circle formula like any other equation. Substitute what you know, then use inverse operations to find $r$ .

Worked example: Radius from circumference

A circular running track has circumference $44\pi$ meters. Find its radius and its area.

\begin{aligned} 2\pi r &= 44\pi \\ r &= \frac{44\pi}{2\pi} = 22 \end{aligned}

The radius is $22$ m. The area is $\pi(22^2) = 484\pi$ square meters.

Worked example: Radius from area

A circle has area $81\pi$ square inches. Find its circumference.

$\pi r^2 = 81\pi$ , so $r^2 = 81$ . The radius is a length, so it's positive: $r = 9$ in. Then $C = 2\pi(9) = 18\pi$ inches.

Parts of circles

A semicircle is half a circle, and a quarter circle is one fourth of a circle. Their areas are that fraction of $\pi r^2$ . For the distance around, take that fraction of the curved part, then add any straight edges.

Worked example: Around a semicircle

A semicircle has diameter $12$ ft. Find its perimeter using $3.14$ for $\pi$ .

The curved part is half the circumference: $\dfrac{1}{2}\pi(12) = 6\pi \approx 18.84$ ft. The straight edge is the diameter, $12$ ft.

Perimeter: $18.84 + 12 = 30.84$ ft. (Exactly, it's $6\pi + 12$ ft.)

Tip

Estimate to check. Since $\pi$ is a bit more than $3$ , a circle's circumference is a bit more than $3$ times its diameter, and its area is a bit more than $3$ times $r^2$ .

Practice

Practice 1

A circle has diameter $14$ cm. What is its circumference in centimeters? Give your answer in terms of $\pi$ .