V = (4/3)πr³ and A = 4πr². A sphere of radius 3 has volume and surface area both 36π, which is a coincidence of that particular radius. The surface is always exactly four times the great circle, and the surface-to-volume ratio is always 3/r.
Four great circles
A sphere’s surface area is exactly four times the area of a great circle — the circle you get by slicing it through the centre.
It is not obvious, and Archimedes proving it around 250 BC was a genuine achievement. There is no easy way to see it: the surface curves in two directions at once, and no simple unrolling argument works the way it does for a cylinder.
It does make a memorable check. The skin of an orange, peeled and flattened, covers exactly four circles of the same radius as the orange.
Cavalieri’s later argument gives a cleaner proof: a sphere’s surface has the same area as the curved surface of the cylinder it fits inside, band for band, because the horizontal stretching and vertical compression cancel exactly.
Why small things cool fast
The surface-to-volume ratio is 3/r. It falls as the sphere grows, and that single fact explains a startling amount of the physical world.
Volume grows as r³ and surface as r², so doubling the radius multiplies contents by eight and skin by four. Big things have proportionally less surface.
Biology. Cells stay small because everything they need crosses the membrane, and a large cell cannot get enough through its relatively tiny surface. It is why cells divide rather than grow, and why anything large evolves lungs, gills or intestinal villi — all devices for adding surface without adding volume.
Heat. Small animals lose heat fast and must eat almost continuously; a shrew starves in hours. Large ones have the opposite problem, and elephants have enormous ears to shed heat they cannot otherwise lose.
Everyday. Crushed ice cools a drink faster than one block of the same mass, because crushing multiplies the surface without changing the volume. The same reasoning explains why flour dust can explode and a sack of flour cannot.
The hemisphere has two areas
Cut a sphere in half and “the surface area” becomes ambiguous, which is why this page reports both.
The curved surface alone is exactly half the sphere’s: 2πr².
The total surface adds the flat circular face, giving 3πr² — three great circles rather than two.
Which you want depends on the object. Painting the outside of a dome needs the curved area. The material to make a solid hemisphere needs the total. Getting the wrong one is a 50% error, and nothing about the arithmetic warns you.
The volume has no such ambiguity: it is exactly half, at (2/3)πr³.
Archimedes and the cylinder
Put a sphere inside the smallest cylinder that contains it. The sphere is exactly two thirds of it — in volume and in total surface area.
Two different quantities, the same ratio. Archimedes considered it his finest result and asked for a sphere-in-cylinder on his gravestone.
He got it. Cicero, serving as quaestor in Sicily around 75 BC, found the tomb overgrown and forgotten, and identified it by that figure. It is one of the few times a mathematician’s own assessment of their best work survives, and one of the few tombs located by a theorem.
This page reports the cylinder’s volume alongside the sphere’s so the ratio is visible rather than asserted, and the test suite checks it at eighty radii.
Where it gets used
Storage and pressure vessels. A sphere holds the most for the least surface, so it needs the least material and handles pressure best. Gas storage tanks are spherical for exactly this reason.
Astronomy. Planets and stars are spherical because gravity pulls equally in every direction, and a sphere is the shape that results.
Chemistry and materials. Reaction rates depend on surface area, so a powdered reagent behaves very differently from the same mass in one lump.
Sport. Ball volumes and surface areas set the aerodynamic behaviour, and the dimples on a golf ball exist to manage the boundary layer over that surface.
Sources and methodology
The sphere formulas are classical results; these are the references.
Method. Every one of the five accepted measurements converts to a radius first, so they cannot disagree with each other — the suite checks all five round-trip back to the same radius across forty sizes. Volumes are separately verified against numerical integration by discs, which is a genuinely independent route rather than the same formula rearranged, and the sphere-to-cylinder ratio is asserted as exactly two thirds in both volume and surface area at eighty radii. That engine is verified on every change against 96 hand-written assertions, including that every input route reconstructs the same sphere, and that the two-thirds cylinder ratio holds in volume and surface area at every radius tested. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The cylinder a sphere sits inside — and is exactly two thirds of — is on the Cylinder Calculator.