How is an ellipsoid's volume calculated?
Volume = (4/3)πabc, where a, b and c are the three semi-axes — half the length along each perpendicular direction. Note these are semi-axes, not full widths: using full dimensions inflates the answer eightfold. When all three are equal it reduces to the sphere formula (4/3)πr³, which is a useful check on your working.
What is the difference between an ellipsoid, a spheroid and a sphere?
An ellipsoid has three different semi-axes. A spheroid has two equal — oblate if it is flattened at the poles like the Earth, prolate if elongated like a rugby ball. A sphere has all three equal. Each is a special case of the one before, so the same volume formula covers all of them.
Why is there no simple formula for surface area?
Because the surface area of a general ellipsoid requires elliptic integrals, which have no elementary closed form. Practical work uses approximations — Knud Thomsen's formula is accurate to about 1.06% for any ellipsoid. This is the same mathematical obstacle that prevents a simple formula for an ellipse's perimeter.
Is the Earth an ellipsoid?
Very nearly. It is an oblate spheroid, flattened by rotation: the equatorial radius is about 6,378 km against a polar radius of 6,357 km, a difference of 21 km. Reference ellipsoids such as WGS 84 model this and underpin GPS. The true shape, the geoid, is bumpier still because gravity varies with the density of the rock beneath.
Where is ellipsoid volume actually used?
Medical imaging estimates organ and tumour volume from three perpendicular measurements on a scan — the standard prostate and bladder volume formulas are exactly this. It is also used for grain and fruit sizing, tank design, and modelling particles in fluid dynamics where shape affects drag.
How accurate is the ellipsoid approximation for real objects?
Good for smoothly rounded shapes, poor for anything lobed or irregular. Clinical use of the formula for organ volume typically carries a known error of 10–20% against true volume measured by segmentation, which is accepted because it is fast and reproducible. Treat the result as an estimate whose accuracy depends on how ellipsoidal the object genuinely is.