the area of a circle

How much pizza is on a 12-inch pizza? How much land does a sprinkler water? These ask for the area enclosed by a circle — the flat region inside the round boundary. The answer, A = pi · r^2, is one of the most famous formulas in all of mathematics.

Here is the intuition without calculus. Slice the disk into very thin pie wedges and lay them out alternately point-up and point-down, like teeth zipping together. The result is almost a rectangle: its height is the radius r, and its long side is half the circumference, pi · r (the other half forms the bottom edge). The area of that rectangle is r times pi · r = pi · r^2, and as the wedges get thinner the shape becomes a perfect rectangle, so the area is exactly pi · r^2. This is the regular-polygon argument (area = half perimeter times apothem) pushed to its limit.

Because the radius is squared, area grows fast: doubling the radius quadruples the area, and tripling it makes the area nine times bigger. A 16-inch pizza is not twice a 8-inch pizza, it is four times the food. Note also that the formula uses the radius, not the diameter — a frequent slip is to plug in the diameter and overshoot by a factor of four.