Ellipsoid Volume Calculator
Find the volume and approximate surface area of an ellipsoid from its three semi-axes.
How it works
V = (4/3)·π·a·b·c · Surface area ≈ 4π · [ ((ab)^p + (ac)^p + (bc)^p) / 3 ]^(1/p), p ≈ 1.6075
An ellipsoid is a stretched sphere, with three independent semi-axes a, b and c along perpendicular directions. Its volume generalises the sphere’s neatly to (4/3)πabc — set all three axes equal to r and it collapses back to (4/3)πr³. Unlike the volume, the surface area has no elementary closed form; the calculator uses the Knud Thomsen approximation, which is accurate to within about 1% for typical shapes.
Worked example
For semi-axes a = 3, b = 4, c = 5: volume = (4/3)·π·(3·4·5) = (4/3)·π·60 = 80π ≈ 251.33 units³. The Knud Thomsen formula gives a surface area of about 199.5 units².
Sources: Ellipsoid — Wolfram MathWorld
Frequently asked questions
Why is the surface area only approximate?
A general ellipsoid’s exact surface area requires elliptic integrals with no elementary closed form. The Knud Thomsen approximation used here is simple and stays within roughly 1% for most axis ratios.
How does this relate to a sphere?
A sphere is the special ellipsoid where a = b = c = r. Substituting gives volume (4/3)πr³ and surface area 4πr², so the sphere formulas are just this calculator with equal axes.
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This calculator is for educational purposes. Double-check important results before acting on them.