Ellipsoid Volume Calculator

An ellipsoid is the three-dimensional generalisation of an ellipse. Where an ellipse is defined by two semi-axes, an ellipsoid has three perpendicular semi-axes — conventionally labelled a, b, and c — that stretch the shape independently along the x, y, and z directions. Common real-world examples include the Earth (an oblate spheroid, slightly flattened at the poles), a rugby ball (a prolate spheroid, elongated along one axis), an egg, a pharmaceutical capsule, and countless engineered components in aerospace, optics, and architecture.

This calculator computes the exact volume and an accurate approximation of the surface area for any ellipsoid. It also works in reverse: give it the volume and any two semi-axes and it will solve for the third. Everything runs locally in your browser — no figures are uploaded or stored anywhere.

How it works

Volume — exact formula

The volume of an ellipsoid is:

V = (4/3) * pi * a * b * c

where a, b, c are the three semi-axes. Because multiplication is commutative the formula is completely symmetric — it does not matter which axis you call a, b, or c. The formula is exact for all ellipsoids, including the two spheroid special cases and the sphere (a = b = c = r).

Solving for an unknown semi-axis is straightforward:

a = (3 * V) / (4 * pi * b * c)

and symmetrically for b and c. The “Solve for” dropdown applies whichever rearrangement you need.

Surface area — Knud Thomsen approximation

Unlike volume, the surface area of a general tri-axial ellipsoid (where all three axes differ) cannot be expressed in a finite number of elementary functions. The exact answer requires elliptic integrals. For a practical calculator, the Knud Thomsen approximation is the standard choice:

S approx 4 * pi * ((a^p * b^p + a^p * c^p + b^p * c^p) / 3)^(1/p)

with p = 1.6075. The maximum relative error is below 1.061% for any combination of axis ratios, making it accurate enough for all engineering and science applications short of precision geodesy. (For the two spheroid special cases exact closed-form expressions do exist; the Thomsen formula still matches them to better than 0.002%.)

SVG diagram

The live diagram shows the equatorial cross-section (a × b plane) as a filled ellipse and the meridional cross-section (a × c plane) as a dashed ellipse overlay, with the three semi-axis arrows labelled with your current values. It rescales automatically so the shape always fits inside the viewport.

Worked example

A storage tank has the shape of an ellipsoid with semi-axes:

• a = 3 m (half-width)
• b = 2 m (half-height)
• c = 1.5 m (half-depth)

Volume:

V = (4/3) * pi * 3 * 2 * 1.5
  = (4/3) * 3.14159265 * 9
  = 37.699 m^3

That is approximately 37,700 litres of capacity.

Surface area (Knud Thomsen, p = 1.6075):

term = (3^p * 2^p + 3^p * 1.5^p + 2^p * 1.5^p) / 3
S = 4 * pi * term^(1/p)
  approx 57.82 m^2

Reverse solve — if the tank volume must be exactly 50 m^3 and b = 2, c = 1.5, what should a be?

a = 3 * 50 / (4 * pi * 2 * 1.5)
  = 150 / 37.6991
  approx 3.979 m

The first row (a = b = c = 3) is a sphere; the formula gives exactly (4/3)pi27 = 113.097.

Formula reference

All calculations use IEEE 754 double-precision floating point (about 15 significant digits), which far exceeds the precision of any physical measurement. pi is used at full double precision internally.