The Area of a Circle

Leila wants to paint a round table top. The tin says one tin covers 1 square metre. Will one tin be enough? She needs the area of the circle. But a circle has no straight sides, so the squares we counted in Area: Counting Squares never fit it exactly. How can we find it?

Cut it into slices

Cut the circle into 8 equal slices, like a pizza. Lay them in a row, one pointing up, the next pointing down, and so on. Nothing is added and nothing is taken away, so the row has the same area as the circle.

A circle cut into 8 equal slices, then the same 8 slices laid in a row, pointing up and down in turn, making a bumpy shape like a parallelogram

The row looks like a parallelogram with bumpy top and bottom edges. Now cut 16 slices, then 32. The thinner the slices, the flatter the bumps, and the more the row looks like a rectangle.

Measure the rectangle

Look at the nearly-rectangle made of many thin slices:

Its height is the length of one slice: the radius of the circle.
Its length is made of the curved edges of half the slices: the other half are along the bottom. So it is half the circumference. In the last lesson we saw that the circumference is 2 × π × radius, so half of it is π × radius.

Many thin slices of a circle laid head to tail, making almost a rectangle. Its height is one radius, and its length is half the way round the circle

The area of a rectangle is length × width, as in The Area of a Rectangle. Here that is (π × radius) × radius. So:

area of a circle = π × radius × radius

A circle with a radius of 10 cm has an area of about 3.14 × 10 × 10 = 314 square centimetres.

Does that make sense?

Put the circle of radius 10 cm inside a square that just fits round it. The square is 20 cm on each side, so its area is 20 × 20 = 400 cm². The circle is smaller than the square: 314 is less than 400. Now look at one small square on the radius, 10 by 10, which is 100 cm². The circle holds a bit more than 3 of those small squares, which matches "π × radius × radius": 3.14 small squares.

A circle of radius 10 cm inside a square 20 cm on each side, divided into four small squares 10 by 10. The circle covers a little more than three of the small squares' worth

Leila's table is 1 metre across, so its radius is half a metre, 0.5 m. Its area is about 3.14 × 0.5 × 0.5, which is about 0.79 m². One tin is enough, with some to spare.

Try it at home

Draw a circle on paper, cut it out, fold it in half three times and cut along the folds to get 8 slices. Lay them head to tail.
Find the area of a round mat with a radius of 2 m.
Double the radius of a circle from 3 cm to 6 cm. How many times bigger does its area get?

Answers: 2. About 3.14 × 2 × 2 = 12.56 m². 3. 3 × 3 = 9 and 6 × 6 = 36, so 4 times bigger.

Words to know

area of a circle - π × radius × radius, found by cutting the circle into thin slices that make almost a rectangle

Big idea

Cutting a circle into thin slices and laying them head to tail makes almost a rectangle, one radius high and half the circumference long. So the area of a circle is π × radius × radius.