Ellipsoid Calculator

Calculate the volume and surface area of an ellipsoid from its three semi-axes. Surface area uses the Knud Thomsen approximation. Free online ellipsoid calculator.

Ellipsoid Properties

Volume

25.1327

cubic units

Surface Area (approx.)

48.9719

square units

Semi-axis a3
Semi-axis b2
Semi-axis c1
Equivalent Sphere Radius1.8171

Semi-Axes

Formulas Used

  • Volume = (4/3) × π × a × b × c = (4/3) × π × 3 × 2 × 1 = 25.1327
  • Surface Area ≈ 4π × ((ab)^p + (ac)^p + (bc)^p) / 3)^(1/p), p = 1.6075 = 48.9719 (approximate)
  • Equivalent sphere radius = ∛(a × b × c) = 1.8171

Note: the surface area above is an approximation (Knud Thomsen's formula, accurate to within ~1.061%). The ellipsoid surface area has no closed-form elementary expression; the exact value requires an elliptic integral.

Quick Answer

To calculate ellipsoid properties with semi-axes a, b, c: Volume = (4/3) x pi x a x b x c. Surface Area is approximately 4 x pi x (((ab)^1.6075 + (ac)^1.6075 + (bc)^1.6075) / 3)^(1/1.6075) — the Knud Thomsen approximation, accurate to about 1%. For a = 3, b = 2, c = 1: Volume = 25.13 cubic units, Surface Area = 48.97 square units (approximate).

Key Facts

  • Formula: Volume = (4/3) x pi x a x b x c (exact)
  • Formula: Surface Area = Knud Thomsen approximation with p = 1.6075 (approximate, ~1% error)
  • The ellipsoid surface area has no closed-form elementary formula
  • An ellipsoid with a = b = c = r is a sphere: volume 4/3 pi r^3, area 4 pi r^2
  • A spheroid has two equal semi-axes (a = b), like a flattened or elongated sphere
  • The Thomsen formula is exact for spheres and most accurate near spherical shapes
  • Earth is an oblate spheroid with semi-axes about 6378 km and 6357 km

Frequently Asked Questions

Use Volume = (4/3) x pi x a x b x c, where a, b, c are the three semi-axes. For a = 3, b = 2, c = 1: Volume = (4/3) x 3.14159 x 6 = 25.13 cubic units. Setting a = b = c = r recovers the sphere formula 4/3 pi r^3.

There is no simple exact formula. Use the Knud Thomsen approximation: SA = 4 x pi x (((ab)^p + (ac)^p + (bc)^p) / 3)^(1/p) with p = 1.6075. It is exact for spheres and has a maximum relative error of about 1.061%.

The exact surface area of a general ellipsoid requires incomplete elliptic integrals, which have no expression in elementary functions. Knud Thomsen's 2004 power-mean formula approximates the value with a simple closed form that is exact for spheres.

A spheroid is an ellipsoid with two equal semi-axes (a = b), produced by rotating an ellipse — flattened (oblate) like Earth or elongated (prolate) like a rugby ball. A general ellipsoid has three different semi-axes.

It is the radius of the sphere with the same volume: r = cube root of (a x b x c), because (4/3) pi r^3 = (4/3) pi abc. For semi-axes 3, 2, 1 the equivalent radius is cube root of 6 = 1.82.

The semi-axes a, b, c are the distances from the center of the ellipsoid to its surface along the three perpendicular principal directions — half the width in each direction. They are the ellipsoid equivalent of a sphere's radius.