Ellipsoid Volume Calculator
An ellipsoid is the 3-D generalisation of an ellipse, defined by three semi-axes a, b and c along mutually perpendicular directions.
Enter Values
How to use this calculator
- Enter the three semi-axes a, b and c — each is half the full length of the ellipsoid along that axis (so for a diameter of 10, enter 5).
- Use the same length unit for all three inputs; the volume comes out in that unit cubed and the surface area in that unit squared.
- Read the volume (and the approximate surface area) instantly — set two or three axes equal to model a spheroid or a sphere.
How it works
An ellipsoid is the 3-D generalisation of an ellipse, defined by three semi-axes a, b and c along mutually perpendicular directions. Its volume is given exactly by V = (4 / 3) x pi x a x b x c. When all three semi-axes are equal (a = b = c = r) this reduces to the familiar sphere volume (4 / 3) x pi x r^3; when two are equal it gives a spheroid (prolate or oblate).
Unlike the volume, an ellipsoid's surface area has no simple closed form, so this tool reports Thomsen's approximation: S is about 4 x pi x [ ((ab)^p + (ac)^p + (bc)^p) / 3 ]^(1/p) with p = 1.6075. This is accurate to within roughly 1.06% for any set of axes and is exact for a sphere. Everything is computed in your browser with standard arithmetic — no data leaves your device.
Worked example
Volume of an ellipsoid with semi-axes 10, 6 and 4. For semi-axes a = 10, b = 6 and c = 4, the volume is (4 / 3) x pi x 10 x 6 x 4 = 1,005.31 cubic units. Because the three semi-axes differ, the shape is a general (tri-axial) ellipsoid rather than a sphere or spheroid.
Common mistakes
- Entering full axis lengths (diameters) instead of semi-axes. The formula uses the half-lengths a, b and c, so a ball of diameter 10 has semi-axis 5, not 10.
- Mixing units between the three inputs (for example a in cm and b in m). Convert everything to one unit first, or the result is meaningless.
- Treating the surface-area figure as exact. It is a very good approximation for a general ellipsoid but only exact when the shape is a true sphere.
Frequently asked questions
What is the formula for the volume of an ellipsoid?
V = (4 / 3) x pi x a x b x c, where a, b and c are the three semi-axes (half-lengths along each perpendicular direction). It is an exact result, not an approximation.
How is this different from a sphere or a spheroid?
A sphere has all three semi-axes equal (a = b = c). A spheroid has exactly two equal (prolate if the odd one is longer, oblate if shorter). A general or tri-axial ellipsoid has all three different. The same volume formula covers every case.
Why is the surface area only approximate?
An ellipsoid's exact surface area involves elliptic integrals with no elementary closed form. This tool uses Thomsen's approximation, which stays within about 1.06% of the true value and is exact when the ellipsoid is a sphere.
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