Why a circle's area is πr²

Why does a circle’s area equal πr²? Cutting it into slices helps us see where the formula comes from.

The radius, called r, measures from the center to the edge. The circumference is the distance around, equal to 2πr.

Cut the circle into equal slices, each meeting at the center. Color half the slices yellow, and the other half blue.

Lay the slices side by side, alternating their directions. Moving these pieces keeps their total area the same.

With more, thinner slices, the curved edges become flatter. As the slices get arbitrarily thin, the shape approaches a rectangle.

Half the circle’s circumference runs along the top, and half along the bottom. Each long side approaches πr, half of 2πr.

Each slice has straight sides of length r, the circle’s radius. As the slices narrow, the rectangle’s height approaches r.

For a radius of two, the rectangle’s width is 2π. Its height is two, giving an area of 4π square units.

A rectangle’s area is its width times its height. So the circle’s area is πr².