The area of a circle

A ratio that depends only on shape. For every square, perimeter² : area is 16 : 1; for every equilateral triangle, 36 : √3. So for circles, C² : A must be fixed too.

Early guesses: the Babylonians found it was about 12, giving A ≈ C²/12. The Egyptians used A ≈ (8d/9)², about 3.16r².

Archimedes (c. 250 BCE) proved the exact rule: area = ½ × circumference × radius = ½ × 2πr × r = πr².

Nīlakaṇṭha's picture. Cut the circle into thin slices, like a pizza, and lay them alternately point-up and point-down. They make an almost-parallelogram:

• base = half the circumference, πr;
• height = r.

Area = πr × r = πr². The thinner the slices, the better the fit.

A sector is a slice between two radii and an arc. By the unitary method again, a sector of θ° has area πr² × θ/360: a semicircle ½πr², a quadrant ¼πr².

A segment is the part between an arc and its chord: sector minus triangle. For 60° and radius r it is πr²/6 – (√3/4)r².