Volume of a Sphere: How Archimedes Found It — and Wanted It on His Tomb

One, two, three

Take a cone, a sphere and a cylinder, all with the same width and the same height. Fill them with water. The volumes come out in the ratio 1 : 2 : 3.

This is the reason the volume of a sphere is 4⁄3 πr³ — found nearly two thousand years before calculus existed. Archimedes was so proud of it that he wanted it on his grave.

Archimedes of Syracuse

Archimedes lived in Syracuse, on Sicily, around 287 to 212 BC. These dates come from later writers. He was famous as an engineer, but his own writings are mathematics. Several of them are letters to mathematicians in Alexandria. One, sent to a man called Dositheus, is called On the Sphere and Cylinder.

The claim

Near the end of its first book, Archimedes proves two things:

Then he draws the consequence: a cylinder that fits exactly around the sphere is one and a half times the sphere. And the same holds for the surfaces — the cylinder's surface, top and bottom included, is one and a half times the surface of the sphere.

Checking it with today's formulas

Archimedes wrote none of these formulas. He compared one shape with another. In today's notation:

cylinder: πr² · 2r = 2πr³     sphere: 4⁄3 πr³ = ⅔ · 2πr³

The surface works the same way: four great circles make 4πr², and the cylinder's surface, top and bottom included, is 2πr · 2r + 2πr² = 6πr² — again one and a half times the sphere's.

A cone with the same base and height is one third of the cylinder (see the volume of a pyramid). So:

cone : sphere : cylinder = ⅔ πr³ : 4⁄3 πr³ : 2πr³ = 1 : 2 : 3

How he found it: a lever

In another letter, called The Method and written to Eratosthenes, Archimedes explains how he found the result.

For this argument, the cone and cylinder are twice as wide as the sphere, but just as tall. He imagined their circular slices as weights on a lever: the slices of the sphere and cone were moved to one fixed point on one side, while the cylinder slices stayed at their different distances on the other. Everything balances — and the balance gives the volume of the sphere.

A lost book under a prayer book

The Method was lost for centuries. A copy made in the 10th century was scraped off and overwritten as a prayer book; the date in that book is 1229.

In 1906, the scholar Johan Ludvig Heiberg recognised the faint text underneath. After the book was sold at auction in 1998, it was imaged with modern methods, and much more of it could be read.

The proof he trusted

The proof in On the Sphere and Cylinder is different. Archimedes fits figures inside the sphere and around it, and shows that the sphere cannot be larger, and cannot be smaller, than his claim. Only one value is left — the same pattern as in his proof for the area of a circle.

He did not start from nothing. In The Method he says that Democritus stated that a cone is one third of its cylinder, and that Eudoxus proved it.

The tomb

According to Plutarch, Archimedes asked his friends to place on his tomb a cylinder enclosing a sphere, with the ratio.

In 75 BC, Cicero says he found the neglected tomb near Syracuse: a small column, overgrown with brambles, bearing a sphere and a cylinder. The local people had forgotten where it was.

Try it yourself

If the cylinder holds 30 litres of water, how much does the sphere hold? And how much the cone?