Ellipsoid Volume Calculator

An ellipsoid is a sphere that has been stretched or squashed along its axes, the smooth three-dimensional shape behind everything from rugby balls and eggs to storage tanks, pressure vessel ends, planets and the way scientists model particles and droplets. It is defined by three semi-axes, the distances from the centre to the surface measured along three directions at right angles to each other, and this calculator works out its volume and surface area from exactly those three numbers. Enter the semi-axes a, b and c and it returns the enclosed volume and an accurate estimate of the surface area, updating instantly as you adjust them. The volume is the headline figure for capacity and material problems, such as the contents of an oval tank, the displacement of an egg-shaped form, or the size of a modelled object, and it comes from a clean, exact formula that is simply the sphere formula with the single radius replaced by the three different semi-axes. The surface area is a little different: unlike the volume, a general ellipsoid has no simple exact surface area formula, so the calculator uses the well established Knud Thomsen approximation, which is accurate to around one percent for the great majority of shapes and is the standard choice when an exact closed form is not available. That makes the tool reliable for real engineering, design and science work, as well as for students meeting ellipsoids for the first time. Because all three axes are independent, you can model anything from a near sphere to a long, thin cigar shape and see how the volume and surface respond. The formulas, the approximation used, and a worked example are set out below.

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Volume is exact. Surface area uses the Knud Thomsen approximation, accurate to about 1% for most ellipsoids.