Why is the area of a circle πr²?

Finding the area of a rectangle is easy: length times width. But a circle has no straight sides at all. So why is its area pi r squared?

First, a quick reminder. The radius, r, goes from the centre to the edge. The distance all the way round is the circumference: two pi r. So half of it is just pi r.

Now cut the circle into eight equal slices, like a pizza. Split it in half, and open both halves out. Then slot them together, like teeth.

Look at this new shape. Its top edge is half the circumference, and so is its bottom edge. But it's bumpy, and a bit slanted.

So let's use thinner slices. Sixteen slices. Sixty four slices. The bumps flatten out, the sides stand up straight, and the shape becomes a rectangle.

Its height is the radius, r. Its width is half the circumference, pi r. So its area is pi r times r: pi r squared! We only moved the slices, so that's the area of the circle too.

Let's try one. A circle has a radius of five centimetres. Square the radius first: five times five is twenty five. Then multiply by pi. The area is about seventy eight point five square centimetres.

So, to sum up. Cut a circle into thin slices, and rearrange them into a rectangle. That rectangle is pi r wide and r tall. So the area of a circle is pi r squared.