Spheres

Archimedes asked for a sphere inside a cylinder to be carved on his tomb, because the sphere has exactly two thirds of the cylinder's volume.

A carving found under the bushes

Why would a mathematician want that picture on his grave? Around 2,200 years ago Archimedes of Syracuse proved two results in his work On the Sphere and Cylinder. The first: the surface area of a sphere is four times the area of its great circle (the largest circle you can cut through it), so A = 4πr². The second: its volume is two thirds of the volume of the cylinder that just fits around it, V = ⅔ × πr² × 2r = ⁴⁄₃ πr³. Plutarch writes that, of all his discoveries, this is the one Archimedes asked his family and friends to place on his grave: a cylinder enclosing a sphere, with the ratio between them written beside it.

An orange peel will not lie flat

Archimedes needed clever arguments because a sphere gives you nothing to unroll. It is the set of all points at the same distance r from a centre, with no flat faces, no edges and no corners. A cylinder or a cone can be cut and laid flat. A sphere cannot, unless you stretch or tear it. Try it with an orange: press the peel flat and it always splits or wrinkles somewhere. Mapmakers run into the same problem from the other side, which is why every flat world map distorts some areas.

The cylinder that fits around the sphere

Press the Archimedes cylinder button in the model. The cylinder has the sphere's radius r and height 2r. Its curved side unrolls into a rectangle 2πr wide and 2r tall, so its area is 4πr², exactly the area of the sphere. Count its two circular ends as well and the whole surface of the cylinder is 6πr²; the sphere's 4πr² is two thirds of that. Surface and volume both come out at two thirds of the cylinder, the pair of results Archimedes liked so much.

Here is a curiosity to check in the model. A sphere of radius 3 has surface 4π × 9 = 36π and volume ⁴⁄₃ π × 27 = 36π, both about 113.1. That happens only at r = 3, because 4πr² = ⁴⁄₃ πr³ only when r = 3. The units still differ (cm² against cm³), so the numbers match while the quantities do not. Larger spheres have the bigger volume number. Smaller ones have the bigger area number.

Half a melon and a 22 cm ball

Cut a sphere in half and you have a hemisphere. Its curved part is half of 4πr², which is 2πr². A solid half, like half a melon, also has a flat circle of area πr², so its whole surface is 3πr². For r = 5 cm that is 75π ≈ 235.6 cm². The volume is half the sphere's, ⅔ πr³. A ball 22 cm across has r = 11 cm. Its surface is 4π × 121 ≈ 1,520.5 cm² and its volume ⁴⁄₃ π × 1,331 ≈ 5,575 cm³, so it holds about 5.6 litres of air.