About the ellipsoid calculator
An ellipsoid is a sphere stretched differently along three perpendicular axes, with semi-axes a, b and c. The volume is simply (4/3)πabc, but the surface area requires elliptic integrals. Most tools fall back on Knud Thomsen’s approximation (up to about 1% error); this calculator evaluates the exact formula using Carlson’s symmetric integrals.
When two semi-axes are equal it is a spheroid: oblate (like Earth) or prolate (like a rugby ball).
Formulas
| Volume | V = (4/3)πabc |
| Surface area | S = 2πc² + (2πab/sin φ)(E(φ,k)·sin²φ + F(φ,k)·cos²φ), cos φ = c/a, computed exactly with Carlson elliptic integrals |
| Thomsen approximation | S ≈ 4π((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p), p ≈ 1.6075 |
Frequently asked questions
Is the ellipsoid surface area here approximate?
No. It is computed from the exact Legendre formula using Carlson elliptic integrals, accurate to about 12 significant digits.