The volume and surface area formulas
V = 4/3 πr³
A = 4πr²
Where the formula comes from
Archimedes proved that a sphere occupies exactly two thirds of the cylinder that just contains it, and he was proud enough of it to have the figure carved on his tomb. That cylinder has volume πr² × 2r = 2πr³, and two thirds of that is 4/3 πr³. The surface, 4πr², is exactly four times the area of the great circle through its middle.
A worked example, step by step
A sphere with a radius of 5 cm
- Radius cubed: 5³ = 125.
- Volume: 4/3 × π × 125 = 523.60 cm³.
- Surface: 4 × π × 5² = 314.16 cm².
- Great circle: 2 × π × 5 = 31.42 cm.
Watch the units: volume goes in cubic units and surface area in square ones. Mixing them is the most common mistake.
Where it turns up in practice
Ball bearings, pressure vessels and storage spheres, where the shape distributes stress evenly and minimises material. In astronomy it gives planetary volumes, and in chemistry it underlies the whole idea of atomic and molecular radii.