Math Core

How far does a bike wheel roll in one turn? How much fencing goes around a round garden? Both questions ask for the distance around a circle, called its circumference. One special number, $\pi$ , connects that distance to the circle's width.

Parts of a circle

Every point on a circle is the same distance from its center.

The radius $r$ is the distance from the center to the circle.
The diameter $d$ is the distance across the circle through the center.

A diameter is made of two radii, so $d = 2r$ and $r = \dfrac{d}{2}$ .

A circle with its center, a radius and a diameter.

Discovering pi

Wrap a string around a can, then compare it with the can's diameter. Try it with a plate, a coin or a tire. Every time, the circumference is a little more than $3$ times the diameter.

That ratio is the same for every circle, no matter how big or small. It's called pi (written $\pi$ ), and it's about $3.14$ . Its digits go on forever without repeating: $3.14159\ldots$ . For quick work you can use $3.14$ or the fraction $\dfrac{22}{7}$ .

Circumference of a circle

\pi = \frac{C}{d} \qquad\text{so}\qquad C = \pi d \quad\text{or}\quad C = 2\pi r

Circumference and diameter are proportional, and $\pi$ is the constant of proportionality.

An answer in terms of $\pi$ keeps the symbol, like $10\pi$ . That's the exact answer. To get a decimal, replace $\pi$ with $3.14$ .

Worked example: Using the diameter or the radius

(a) A circle has diameter $10$ cm. Find its circumference in terms of $\pi$ , then approximate it using $3.14$ .

$C = \pi d = 10\pi$ cm, which is about $10 \times 3.14 = 31.4$ cm.

(b) A circle has radius $7$ inches. Find its circumference using $\dfrac{22}{7}$ for $\pi$ .

$C = 2\pi r \approx 2 \times \dfrac{22}{7} \times 7 = 44$ inches.

Working backward

If you know the circumference, divide by $\pi$ to get the diameter.

Worked example: From circumference to radius

A round rug has circumference $50.24$ feet. Using $3.14$ for $\pi$ , find its radius.

\begin{aligned} C &= \pi d \\ 50.24 &= 3.14d \\ d &= 50.24 \div 3.14 = 16 \end{aligned}

The diameter is $16$ feet, so the radius is $16 \div 2 = 8$ feet.

Half circles

The distance around a semicircle (half a circle) includes the curved part and the straight diameter.

A semicircle with diameter 8 cm.

Worked example: Perimeter of a semicircle

Find the perimeter of the semicircle above, using $3.14$ for $\pi$ .

The curved part is half the circumference of a circle with diameter $8$ : $\dfrac{1}{2} \times 8\pi = 4\pi \approx 12.56$ cm.

Add the straight side: $12.56 + 8 = 20.56$ cm.

Common mistake

Check whether you were given the radius or the diameter. With radius $5$ , the circumference is $2\pi(5) = 10\pi$ , not $5\pi$ . If you use $C = \pi d$ , make sure $d$ really is the diameter.

Tip

Quick check: the circumference should be a bit more than $3$ times the diameter. A circle $10$ cm across should have a circumference a little over $30$ cm.