Spheres and hemispheres

A sphere is all the points in space at distance r from a centre O. Cut it through O and you get two hemispheres.

Archimedes' discovery: a sphere's surface area equals the curved surface of the cylinder that fits tightly round it: 2πr × 2r = 4πr². Try it with string: the string that covers a ball exactly fills 4 circles of the same radius.

This is why a rectangular world map can show the whole globe (though it stretches shapes near the poles).

Volume: (4/3)πr³. Picture the sphere made of many thin cones, each from the centre out to the surface, all of height r. Together their bases cover the surface 4πr², so the volume is (1/3) × 4πr² × r.

The sphere fills 2/3 of its enclosing cylinder: (4/3)πr³ ÷ 2πr³ = 2/3.

Hemisphere:

• curved surface 2πr²;
• total surface 3πr² (adding the flat circle πr²);
• volume (2/3)πr³.