One Formula Covers the Sphere, the Egg and the Planet

Stretching one axis stretches the volume in proportion

Because the three semi-axes multiply, each one is linear in the answer: making a shape half as long in one direction halves the volume, and doing it in all three directions divides the volume by eight. That is a friendlier relationship than the cube law that governs scaling a whole object, and it is why an ellipsoid is the natural model whenever something is squashed in one direction only — a red blood cell, a lens, a squashed planet. It also means the three numbers can be reordered freely: 5, 3, 4 and 3, 4, 5 describe differently oriented shapes with identical volumes, which is occasionally a useful check on whether an input has been entered in the wrong box.

Why a planet is not a sphere

Rotation throws material outward at the equator and not at the poles, so any spinning fluid body settles into a flattened ellipsoid. Earth's flattening is small — about one part in 298, or twenty-one kilometres of difference between the two radii — and it is enough to matter for navigation, for satellite orbits and for the definition of the coordinate systems every map uses, which are built on a reference ellipsoid rather than a sphere. Jupiter, which turns once in under ten hours, is visibly squashed in a small telescope. The rule is general: the faster the spin and the less rigid the body, the further from a sphere it settles.

Frequently asked questions

Is the surface area as simple?

Not remotely. The volume has an exact elementary formula and the surface area of a general ellipsoid does not — it needs elliptic integrals, and the figures you see quoted come from approximations rather than from an exact expression. The tool's area is one of those approximations and is accurate to a fraction of a per cent for shapes that are not extremely elongated, which is enough for anything practical and not enough to quote to five figures. The sphere is the special case where everything is clean, and it is one of several reasons why so much of physics is done on spheres.

How do I measure the semi-axes of a real object?

Take the three greatest widths at right angles to each other and halve each one. Callipers do this well; a tape does not, because it follows the surface rather than crossing it, and the difference on a rounded object is exactly the error you are trying to avoid. The result is only as good as the assumption that the object really is an ellipsoid — an egg is noticeably not one, being fatter at one end, so treating it as a symmetric shape overstates the volume slightly. When accuracy matters more than convenience, displacing water measures the volume directly and skips the geometry entirely.