Ellipsoid — Volume + Surface Area
Ellipsoid: V = (4/3)π·a·b·c (exact), SA ≈ Knud Thomsen 1.6075 approximation (≤1.061% error worldwide).
Result
- Semi-axis a3
- Semi-axis b4
- Semi-axis c5
- Volume ((4/3)πabc)251.327412
- Surface area (Thomsen)199.501704
- Sphere checkno — true ellipsoid
Step-by-step
- V = (4/3)π × a × b × c = (4/3)π × 3 × 4 × 5 = 251.3274. (Exact.)
- Knud Thomsen approximation, p = 1.6075:
- Inner = (aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3 = 85.1499.
- SA ≈ 4π × Inner^(1/p) = 4π × 15.8758 = 199.5017.
How to use this calculator
- Enter the three semi-axes a, b, c.
- Read the exact volume and the Thomsen-approximation surface area.
- For a = b = c the result equals the sphere formulas exactly.
About this calculator
An ellipsoid is a sphere stretched along three perpendicular axes (semi-axes a, b, c). Volume V = (4/3)πabc — exact and elegant (when a = b = c the formula reduces to the sphere's (4/3)πr³). Surface area has no closed-form expression in elementary functions (it requires elliptic integrals). The Knud Thomsen (2004) approximation SA ≈ 4π × ((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p) with p ≈ 1.6075 is accurate worldwide to ≤1.061% relative error and reduces exactly to 4πr² for a sphere. Source: Knud Thomsen, "Surface area of an ellipsoid" (NumericAnalysis.com, 2004); Mathematica documentation. Earth is a slightly oblate ellipsoid (a = b = 6378.137 km, c = 6356.752 km — WGS84).