Geometry · A visual proof
Why is the area of a circle πr²?
Cut a circle into thin wedges, then slide them back together a different way. The pieces rearrange into a shape you already know how to measure — a rectangle.
The reasoning
Slice the circle
Cut it from the centre into n identical wedges, like a pizza. Each wedge has two straight sides (each one a radius, length r) and one curved crust (a small piece of the circumference).
Alternate the wedges
Lay them in a row, flipping every other one so the tips point up, down, up, down. They mesh together into one long strip — almost a rectangle.
Where the width comes from
Add up every curved crust and you get the whole way around the circle — the circumference, 2πr. After alternating, half of that crust lines the top of the strip and half lines the bottom. So each long edge is half the circumference:
Where the height comes from
The straight sides of the wedges become the short sides of the rectangle. Each one is a radius, so the height is exactly r.
Cutting and rearranging never changes the area, so this is the circle's area. As the slices get thinner the jagged edges flatten into a perfect rectangle, and the “≈” becomes “=”.