Circumference and Area of a Circle

The parts of a circle

Wheels, coins, plates, clock faces, the base of a mug — circles turn up everywhere. Two everyday questions follow them around: how far is it round the edge, and how much space fits inside? This page shows how to answer both from a single measurement of the circle.

A circle is the set of all points the same distance from one middle point. That middle point is the centre. The distance from the centre out to the edge is the radius (we write it r). A straight line right across the circle, through the centre, is the diameter (d).

The diameter is just two radii laid end to end, so the diameter is always twice the radius:

d = 2r

So a wheel with a radius of 30 cm has a diameter of 60 cm straight across, and a plate with a diameter of 20 cm has a radius of 10 cm.

The number π

Here is the magic. Take any circle, measure all the way around the edge (its circumference), and divide by the diameter straight across. You always get the same number — a little bit more than 3. We call it π ("pi"), and it is about 3.14:

\pi \approx 3.14

Another way to picture \pi: it tells you how many diameters fit around the edge of the circle. You can lay a little more than three diameters around the rim of every circle, big or small. For rough work people sometimes use the fraction \tfrac{22}{7}, which is also close to 3.14.

No! Its digits go 3.14159265\ldots forever, never settling into a repeating pattern. Computers have worked out trillions of digits and still no end is in sight. For everyday school work, 3.14 (or \tfrac{22}{7}) is plenty close.

Circumference: the distance round the edge

Because \pi is the circumference divided by the diameter, the circumference is just \pi times the diameter:

C = \pi d

And since the diameter is twice the radius (d = 2r), we can also write it with the radius. This is exactly the same formula:

C = 2\pi r

Worked example. A bicycle wheel has a radius of 30 cm. How far does it roll in one full turn?

C = 2\pi r \approx 2 \times 3.14 \times 30 = 188.4 \text{ cm}

So one turn of the wheel carries the bike about 188 cm — nearly two metres — forward.

Roll a round wheel along the ground for exactly one full turn. The distance it travels is its circumference — once round the rim equals once along the floor. That is why big wheels cover more ground per turn than little ones: a bigger circle has a longer way around.

Area: the space inside

The area is the amount of space inside the circle — how much paint it would take to colour it in. It depends on the radius squared:

A = \pi r^2

Remember r^2 means r \times r — do that first, then multiply by \pi.

Worked example. A round pizza has a diameter of 30 cm. How much pizza is there? First halve the diameter to get the radius: r = 15 cm. Then:

A = \pi r^2 \approx 3.14 \times 15^2 = 3.14 \times 225 = 706.5 \text{ cm}^2

Another example. A clock face has a radius of 10 cm, so its area is \pi \times 10^2 = 100\pi \approx 314 \text{ cm}^2.

round pizza A pizza is a circle, and its area is how much there is to eat. Double the radius and you do not get twice the pizza — you get four times as much, because the radius is squared. A 30 cm pizza has four times the food of a 15 cm one, even though it is only twice as wide!

The two formulas look alike, so they are easy to mix up:

The circumference uses the diameter (or 2r): C = \pi d = 2\pi r.
The area uses the radius squared: A = \pi r^2. Don't swap them!
If a question gives you the diameter, remember the radius is half of it (r = \tfrac{d}{2}) before you use A = \pi r^2.

Try it: a fresh circle each time

Here is a circle with a random radius. Its radius and diameter are marked, and its circumference and area are worked out underneath using \pi \approx 3.14. Press Refresh for a brand-new circle and follow the sums.

See both on one circle