Area of a Circle: How Archimedes Proved It with a Triangle

A circle as a triangle

Why is the area of a circle πr²? Long before that formula was written, Archimedes gave a geometric answer — a derivation without calculus: a circle has the same area as one particular triangle.

The Measurement of a Circle

Around 250 BC, Archimedes wrote a short work called the Measurement of a Circle. What survives is probably only part of a longer book. Its first proposition is the one we want:

A curved shape and a straight one, with exactly the same area.

A visual argument

Imagine cutting the circle into thinner and thinner rings and straightening each one. The outer ring is as long as the circumference, the inner ones are shorter, down to a point at the centre. Stacked, they form a triangle whose base approaches the circumference and whose height is the radius.

Polygons inside the circle

Archimedes worked with polygons. A regular polygon can be cut into triangles from its centre, so its area is

½ · perimeter · (distance from the centre to a side)

Double the number of sides again and again, and the polygon hugs the circle more and more closely. The gap between the two can be made smaller than any area you choose.

Two impossibilities

Call the triangle K (legs r and C).

1. Suppose the circle were larger than K. Then an inscribed polygon can be made larger than K too. But its perimeter is shorter than the circumference, and its distance to the sides is shorter than the radius — so its area is smaller than ½ · r · C = K. Impossible.
2. Suppose the circle were smaller than K. Then a polygon drawn around the circle can be made smaller than K. But its distance to the sides is exactly the radius, and its perimeter is longer than the circumference — a result Archimedes establishes at the beginning of On the Sphere and Cylinder. So its area is larger than K. Impossible again.

The circle is neither larger nor smaller than the triangle. It is equal. In today's notation:

½ · r · 2πr = πr²

Archimedes used the same squeeze from inside and outside for a harder shape: the volume of a sphere.

Pi between two fractions

The same short book goes further. With polygons of 96 sides, inside and outside the circle, Archimedes showed that the circumference lies between 3 10⁄71 and 3 1⁄7 times the diameter.

3 10⁄71 ≈ 3.1408    and    3 1⁄7 = 22⁄7 ≈ 3.1429

Three and one seventh is 22⁄7, the value many people still use for π. It is the upper bound Archimedes obtained from the outer polygon — not an exact value of π.

Try it yourself

Using 22⁄7 as an approximation, a circle of radius 7 has a circumference of about 44. What area does Archimedes' triangle give? And for a circle of radius 14?