How to find the circumference of a circle

The circumference of a circle we can find by multiplying pi ($\pi$) by the diameter of the circle:

$$C = \pi \times d$$

The diameter of a circle is actually the same as the radius times 2. It is twice as long as the radius. So we can also write the formula as

$$C = \pi \times 2 \times r$$

which you will also see written as $2\pi r$. Same formula, same numbers, just in a different order. The one thing to watch is that this formula wants the diameter, whereas the area formula wants the radius.

When do I use this?

When you want the distance all the way round a circle, its perimeter. You need either the diameter or the radius. What those are, and how they relate, is in circles.

The circumference is a length, so it comes out in cm or m, not in square units. That's one way to tell it apart from the area.

The procedure

1. Find the diameter. If you have the radius, multiply it by 2.
2. Put the diameter into $C = \pi \times d$.
3. Work it out on the calculator.

Say we have a circle with a radius of 3. To find the circumference, we use the circumference formula, but here we have to use the diameter. As we can see from the figure, the diameter is twice as long as the radius:

So we multiply our radius by 2 to find the diameter:

$$3 \times 2 = 6$$

Now we can put our diameter into the formula for the circumference:

$$\pi \times 6 \approx 18.85$$

So a circle with a radius of 3 has a circumference of about 18.85. Together with the area, which came to about 28.27, we now have both numbers for the same circle.

Worked examples

Example 1: the radius is 3

This is the one above. Double the radius to get the diameter, then multiply by $\pi$:

$$3 \times 2 = 6, \quad \pi \times 6 \approx 18.85$$

Example 2: the diameter is given

A circle has a diameter of 10. Then there's nothing to convert. We put 10 straight in $d$'s place:

$$\pi \times 10 \approx 31.42$$

So the circumference is about 31.42. If the diameter was 10 cm, the circumference is about 31.42 cm.

Common mistakes

• "I put the radius straight into π × d." The formula wants the diameter. With a radius of 3, that's $\pi \times 6$, not $\pi \times 3$. Double the radius first, or use $\pi \times 2 \times r$.
• "I mix up the circumference and the area." The circumference is all the way round, $\pi \times d$, and it's a length. The area is what's inside, $\pi \times r^2$, and it's in square units. Different formula, different part of the circle, different unit.

Related

The parts of the circle are in circles. The other formula for the same circle is in how to find the area of a circle, and it can be turned round to give the radius: finding the radius of a circle from its area.