Circumference and Diameter

The radius

Every point on a circle is the same distance from its center. A straight line from the center out to the circle is called a radius, and its length is written r.

The radius runs from the center to the edge of the circle.

The diameter

A straight line all the way across the circle, through its center, is called a diameter, and its length is written d. A diameter is made of two radii end to end, one on each side of the center, so it is always twice as long as the radius: d = 2r.

A circle with a radius of 7 cm has a diameter of 2 × 7 = 14 cm. Going the other way, the radius is half the diameter: a circle with a diameter of 10 cm has a radius of 10 ÷ 2 = 5 cm.

The diameter crosses the whole circle through its center: two radii end to end.

The circumference, and pi

The distance all the way round the edge of a circle is called its circumference. Wrap a piece of string once around a can, then lay the string straight beside the width of the can. The string is a little more than 3 times as long as the width. Do the same with a plate, a coin or a bicycle wheel, and the answer is the same every time: the circumference is a little more than 3 diameters.

This number is called pi, and it is written $\pi$π. It never changes, because every circle is the same shape: a bigger circle is a smaller circle scaled up, so doubling the diameter doubles the circumference too. Pi is 3.14159… and its digits go on forever, so in working we use 3.14 or the fraction $\frac{22}{7}$22/7, which are both close to it.

A wheel with a diameter of 1 rolls along a line. After one full turn, the mark on its rim touches the line again at 3.14159…, which is $\pi$π. The length of line it has rolled over is its circumference.

$C=\pi d$C = πd, and $C=2\pi r$C = 2πr

So the circumference is exactly $\pi$π diameters: $C=\pi d$C = πd. The diameter is 2r, so the circumference is also $C=2\pi r$C = 2πr. Use $C=\pi d$C = πd when you know the diameter, and $C=2\pi r$C = 2πr when you know the radius.

A circle with a diameter of 10 cm has a circumference of $\pi \times 10$π × 10, which is about 3.14 × 10 = 31.4 cm. A circle with a radius of 7 cm has a diameter of 14 cm, so its circumference is about $\frac{22}{7}\times 14=44$22/7 × 14 = 44 cm.

Check the size of the answer: the circumference should be a little more than 3 times the diameter. For the diameter of 14 cm, 3 × 14 = 42, and 44 is a little more than 42.

The circumference, the whole way round the edge, is $\pi$π times the diameter d.

The usual mistakes

Using the radius as the diameter. $C=\pi d$C = πd needs the distance all the way across. For a radius of 7 cm, $\pi \times 7$π × 7 is only half of the circumference. Double the radius first: $\pi \times 14$π × 14.

Squaring the radius. r × r belongs to the area of a circle, the space inside it. The circumference is a length round the edge: $\pi$π times the diameter, which is $2\pi r$2πr.