Why a circle’s area is pi r squared

Almost everybody remembers pi r squared. Far fewer get full marks on circle questions.

The formula is not the hard part. The hard part is knowing what the letters mean, and spotting when the question has quietly handed you something else.

Where the formula comes from

You do not have to take pi r squared on trust. Here is the picture that explains it.

Take a circle and cut it into a lot of thin slices, like a pizza. Now lay the slices out in a row, alternating point up and point down. The shape you get looks almost like a rectangle, and it looks more like one the thinner you cut the slices.

How long is that rectangle? Half the slices point up and half point down, so the top edge is made of half the circle’s outer edge. The whole outer edge is the circumference, 2 pi r. Half of that is pi r.

How tall is it? Each slice reaches from the centre to the edge, so the height is the radius, r.

Area of a rectangle is length times height. So the area is pi r times r, which is pi r squared.

That is the whole derivation. It is worth doing once with scissors and paper, because after that the formula is a memory of a shape rather than a string of symbols.

Mistake 1: using the diameter

This is the big one.

The letter r is the radius: centre to edge. The diameter is edge to edge, straight through the centre, so it is twice the radius.

Exam questions often give the diameter, because it is the easier thing to measure. If you drop the diameter into pi r squared, your answer is four times too big, because you squared something twice as large.

Read the question twice. If it says diameter, or “the circle is 10 cm across”, halve it first.

Mistake 2: squaring the wrong thing

Pi r squared means pi times (r squared). The squaring applies to the radius only.

Square the radius first. Then multiply by pi.

If you type the whole thing as (pi r) squared, you have multiplied by pi twice, and your answer is about 3.14 times too big.

Mistake 3: mixing up area and circumference

Two formulas look alike and are used for different jobs:

• Circumference = 2 pi r. The distance around the edge. Measured in cm, m, inches.
• Area = pi r squared. The space inside. Measured in cm squared, m squared, square inches.

The units tell you which one the question wants. If the answer should be in square centimetres, you need the area. If it should be in centimetres, you need the circumference.

This is also the quickest way to check your own work. An area answer with no “squared” in the units is wrong somewhere.

Mistake 4: rounding too early

Do not replace pi with 3.14 at the start and then carry that through three more steps. The error grows each time.

Keep pi as pi for as long as you can, use the pi button on the calculator, and round only on the final line. If the question asks for an exact answer, leave pi in it: write 25 pi, not 78.5.

A worked example

A circular table top has a diameter of 1.4 metres. What is its area, to two decimal places?

1. The question gives the diameter, 1.4 m. Halve it: the radius is 0.7 m.
2. Square the radius: 0.7 times 0.7 is 0.49.
3. Multiply by pi: 0.49 times pi is about 1.5394.
4. Round: 1.54 square metres.

Check the units. Metres squared, which is right for an area.

Now see how it goes wrong. Use 1.4 instead of 0.7 and you get 6.16 square metres, four times too much. That one misread costs the whole question.

Half circles and rings

Once the formula is solid, the usual exam twists are easy.

A semicircle is half a circle, so work out pi r squared and halve it. A quarter circle is a quarter, so divide by four.

A ring, or annulus, is the big circle minus the small circle. Work out both areas and subtract. Do not subtract the radii and then square the result, which is a different and wrong number.

How to practise it

Do not practise the formula. You already know the formula. Practise the reading.

Write out ten questions where some give a radius and some give a diameter, mixed up. For each one, write down only the radius you will use. Do not calculate anything. Check all ten, then do ten more.

That drills the step where the marks actually go.

FutureSchool teaches circle area in its Maths programme as three separate pieces: a video lesson on where the formula comes from, a worksheet on reading radius from diameter, then a test. Circles appear in the upper primary years, again in lower secondary, and again in GCSE revision.

If you are not sure which topics your child has secure, the free assessment shows the gaps in about twenty minutes.

Questions parents ask

What is the formula for the area of a circle?

Area equals pi times the radius squared. Written out: A = pi r squared. The radius is the distance from the centre to the edge, which is half the diameter.

Is it pi r squared or 2 pi r?

Pi r squared is the area, measured in square units. 2 pi r is the circumference, the distance around the edge, measured in ordinary units. If the answer should be in square centimetres, you want pi r squared.

Do I square the radius or multiply by pi first?

Square the radius first, then multiply by pi. Squaring pi r as a whole gives a completely different number.

What do I do when the question gives me the diameter?

Halve it to get the radius, then use the formula. Putting the diameter straight into pi r squared makes the answer four times too big.