Ellipsoid Volume Calculator

An ellipsoid is a three-dimensional surface obtained by stretching or compressing a sphere along three mutually perpendicular directions. The distances from its center to the surface along those directions are its semi-axes, usually named a, b, and c. When all three are equal, the shape is a sphere. When two are equal, the shape is a spheroid. When all are different, it is often called a triaxial ellipsoid. Three positive semi-axis lengths completely determine the volume and geometric proportions of the idealized shape. The ellipsoid volume formula is exact. It multiplies four thirds of pi by the product of the three semi-axes. This follows naturally from scaling a unit sphere independently in three directions: every scale factor multiplies the enclosed volume. Doubling only one axis doubles the volume, while doubling all three axes multiplies volume by eight. Because the inputs are radii rather than full diameters, divide measured diameters by two before entering them. If inputs use centimeters, for example, the result is cubic centimeters. Unlike volume, the surface area of a general triaxial ellipsoid has no simple elementary formula. Exact expressions involve elliptic integrals. This calculator uses Knud Thomsen's approximation with exponent 1.6075, a widely used formula that blends the three pairwise axis products. It is exact for a sphere and typically very accurate for practical shapes, though it should still be treated as an approximation for highly elongated or flattened ellipsoids. The displayed note distinguishes this value from the exact volume. Ellipsoid models are useful in geometry, astronomy, geodesy, engineering, medicine, manufacturing, and estimates involving tanks, particles, organs, planets, or rounded objects. Real objects may not be perfect ellipsoids, so measurement uncertainty can matter more than numerical rounding. Measure all three full widths through the center, halve each width, and use a consistent unit. The surface result uses square units and volume uses cubic units. A useful check is that permuting a, b, and c cannot change either answer. For a sphere, entering the same radius three times should reproduce the familiar sphere formulas. Results are rounded to six decimal places for readability while the underlying calculation retains more precision.