Ellipsoid Volume Calculator
An ellipsoid is a three-dimensional shape where every cross-section is an ellipse. Think of it as a stretched or squashed sphere: a rugby ball, the Earth itself (slightly flattened at the poles), or an egg-like form are all real-world examples. To describe one precisely you need three semi-axes, labelled a, b and c, each measured from the centre of the solid to a surface point along a different perpendicular direction. The volume formula is exact: V = (4/3) times pi times a times b times c. When all three semi-axes are equal it collapses to the familiar sphere formula. Surface area is a different story. Unlike the sphere there is no tidy closed-form expression for a general ellipsoid, so this calculator uses the Knud Thomsen approximation, which is accurate to within about 1.1 percent for any shape of ellipsoid. You enter the three semi-axes in any consistent unit, metres, centimetres, millimetres or anything else, and the calculator returns the volume and the approximate surface area in the corresponding cubic and square units. The ratio of each pair of semi-axes is also shown, which is useful for describing how elongated or flattened the shape is. The tool is useful for students studying geometry, engineers modelling storage tanks or biological cells, and anyone working with three-dimensional shapes that are not simple spheres. If your three semi-axes are all equal, the result will match the sphere volume formula exactly.
Surface area uses the Knud Thomsen approximation (error less than 1.1%). All units must be consistent: if axes are in cm, volume is in cm cubed and area in cm squared.
How it works
The volume formula is V = (4/3) π a b c, where a, b and c are the three semi-axes. This is exact for any ellipsoid. The Knud Thomsen surface area approximation is SA ≈ 4π × ((apbp + apcp + bpcp) / 3)1/p where p = 1.6075. This form was chosen because it minimises the maximum relative error over all possible axis ratios, keeping the error below 1.061 percent. The axis ratios a:b, a:c and b:c describe the shape: values close to 1 mean nearly equal axes and a near-sphere; larger values indicate a more elongated or flattened form.
Worked example
With semi-axes a = 5, b = 4, c = 3: volume = (4/3) × π × 5 × 4 × 3 = 251.33 cubic units. The Knud Thomsen approximation gives a surface area of approximately 199.50 square units. The axis ratios are a:b = 1.25, a:c = 1.67, b:c = 1.33, showing a moderately elongated shape that is widest along the a-axis.
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