Why is the area of a circle πr²?

It is easy to see why the area of a rectangle is height × width. It is less obvious why the area of a circle is πr². To explain it we need a pizza cut into infinitely many slices. But first, three terms:

The explanation

Imagine a pizza. You have many friends, and split it into many pieces. Unfortunately, your friends found out that there is pineapple on your pizza, and never came. Nonetheless, you don't mind, pineapple people be pineappling. So you conduct an experiment.

You take one half of the pizza and interlock it with the other, with every other slice pointing up and down, as shown below. The more pieces, the more the shape looks like a rectangle. Drag the slider and see how the pieces more and more look like a rectangle.

With an infinite number of pieces, the sides even out, or flatten, into a perfect rectangle. You can see this already with 16 or 32 pieces.

Which pizza should you buy?

Double the radius and the area becomes four times larger, not twice. A fair price for a pizza with twice the radius is therefore four times the price, not two. Or is it? Many don't really enjoy the crust. And the crust is often about the same width, regardless of the size of the pizza. Let us define the width of the crust as c. Then the area of the "fill", the interesting non-crusty part of the pizza, is:

Let's compare two pizzas. Pizza B is seemingly 4 times as large.

The formula tells us two things:

• When the crust c = 0, then Pizza B's fill is 4 times as large
• When the crust c increases towards r, then Pizza B's fill is nearing infinetly times as large (as Pizza A is crust)

Compare the two pizzas below. Pizza A has a radius of 20 cm, Pizza B has a radius of 40 cm, and the crust is 5 cm wide.

Pizza B's fill is 5,44 times as large as Pizza A's, even though the radius is double.