Ellipsoid Calculator

An ellipsoid is a 3D surface shaped like a stretched or squashed sphere, where all cross-sections are ellipses. It is defined by three semi-axes a, b, and c along the x, y, and z axes respectively. A sphere is a special case where a = b = c.

The volume of an ellipsoid is calculated using the formula V = (4/3) × π × a × b × c, where a, b, and c are the three semi-axes. For example, if a = 3 cm, b = 6 cm, and c = 8 cm, the volume is (4/3) × π × 3 × 6 × 8 ≈ 603.19 cm³.

Unlike volume, the surface area of a general ellipsoid has no exact elementary formula. This calculator uses Thomsen's approximation: SA ≈ 4π × ((a^p × b^p + a^p × c^p + b^p × c^p) / 3)^(1/p), where p = 1.6075. This approximation is accurate to within about 1% for most ellipsoids.

Using the formula V = (4/3) × π × a × b × c with a = 3, b = 6, c = 8: V = (4/3) × π × 3 × 6 × 8 ≈ 603.19 cm³.

No. Volume is always a non-negative quantity. Since volume depends on the product of the three semi-axes (all of which must be positive lengths), the result is always a positive number.

Not necessarily. If an ellipsoid's semi-axes are larger than the radius of a given sphere, the ellipsoid can have a greater volume. However, among all ellipsoids with the same surface area, the sphere has the maximum volume.

A spheroid is a special ellipsoid where two of the three axes are equal. A prolate spheroid (like a rugby ball) has one axis longer than the other two equal axes. An oblate spheroid (like the Earth) has one axis shorter than the other two equal axes.

Ellipsoid volume calculations are widely used in medicine (estimating organ volumes from ultrasound measurements such as kidneys, liver, or tumors), engineering (tank and vessel design), astronomy (modeling planet shapes), and food science (estimating volumes of egg-shaped objects).