How to find the area of a circle

The area of a circle we can find by multiplying pi ($\pi$) by the radius of the circle squared:

$$A = \pi \times r^2$$

It is one formula, and the only thing you have to keep straight is that it wants the radius, not the diameter.

When do I use this?

Whenever you know the radius of a circle and want to know how much there is inside it. If you've been given the diameter instead, remember from circles that the diameter is the radius times 2, so you halve it to get the radius first.

The area comes out in square units, cm² or m², because we're multiplying a length by a length. The unit is part of the calculation, which is explained in units.

The procedure

1. Find the radius $r$.
2. Put $r$ into the formula, in $r$'s place.
3. Work it out on the calculator.

Say we have a circle with a radius of 3:

We use the formula for the area and put our radius in $r$'s place:

$$\pi \times 3^2 \approx 28.27$$

So the area of a circle with a radius of 3 is about 28.27.

The line $\pi \times 3^2$ is itself an exact answer. Since $3^2 = 9$, it can also be written as $9\pi$. The calculator only turns it into a decimal, about 28.27. If the radius was 3 cm, the area is about 28.27 cm².

Worked examples

Example 1: the radius is 3

This is the one above. Radius in $r$'s place, square it, multiply by $\pi$:

$$\pi \times 3^2 \approx 28.27$$

Example 2: the diameter is given

A circle has a diameter of 10. The area formula wants the radius, and the diameter is twice the radius, so the radius is

$$10 \div 2 = 5$$

Now we can put 5 in $r$'s place:

$$\pi \times 5^2 \approx 78.54$$

So the area is about 78.54.

We can check that the first step was needed: if we had put the diameter straight into the formula, we would have got $\pi \times 10^2 \approx 314.16$, four times too much, because $10^2$ is four times $5^2$.

Common mistakes

• "I forget to square the radius." The formula is $\pi$ times $r$ squared, and $r^2$ means $r \times r$. With a radius of 3 that's 9, not 3, so the area is $\pi \times 9$, not $\pi \times 3$.
• "I put the diameter into the area formula." The area formula wants the radius. Halve the diameter first. It's the circumference formula that wants the diameter, and that's the other guide: how to find the circumference of a circle.
• "I give the area in cm instead of cm²." Area is a length times a length, so the unit gets multiplied too. If the unit on your answer isn't a square unit, something has gone wrong.

Related

The names of the parts of a circle are in circles. If you have the area and need the radius, the formula can be turned around: finding the radius of a circle from its area.