Why is there no simple formula for the surface area of an ellipsoid?

The surface area of a general ellipsoid with three different semi-axes cannot be expressed with elementary functions, it leads to so-called elliptic integrals. That is why an accurate approximation is used in practice, for example Thomsen's formula with an error of up to roughly 1.1 percent.

Where does an ellipsoid appear in practice?

The Earth has the shape of an oblate spheroid, being slightly flattened at the poles. Ellipsoids are also used in geodesy as a reference body, in optics, in the modelling of tanks and in biology when estimating the volume of cells or organs.

Ellipsoid: volume and surface area - online calculation, formula

Q: What is the formula for the volume of an ellipsoid?

The volume of an ellipsoid is calculated as V = 4/3 · π · a · b · c, where a, b and c are the lengths of the three semi-axes. If all three are equal, the formula turns into the volume of a sphere 4/3 · π · r cubed.

Q: What is a spheroid?

A spheroid, or ellipsoid of revolution, arises by rotating an ellipse about one of its axes, so two semi-axes are equal. If the third one is shorter, the solid is oblate, if longer, it is prolate. An exact surface area formula exists for both cases.

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  Ellipsoid - surface area and volume

Interactive calculator — drag the end points of the semi-axes or type in the values and you see the volume and the surface area right away.

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Formula & notes

V = \frac{4}{3} π \cdot a \cdot b \cdot c

S \approx 4 π {(\frac{a^{p} b^{p} + a^{p} c^{p} + b^{p} c^{p}}{3})}^{\frac{1}{p}}

V = volume of the ellipsoid

S = surface area of the ellipsoid

a, b, c = lengths of the three semi-axes

p = 1.6075 (Thomsen constant)

the volume is exact, the surface area with three different semi-axes is approximate

What is it?

An ellipsoid is a solid every plane cross-section of which is an ellipse (or a circle). It is determined by three mutually perpendicular semi-axes a, b, c. If all three are equal, the solid is a sphere; if two of them are equal, the result is a spheroid (an ellipsoid of revolution) — oblate, like our Earth, or prolate, like a rugby ball.

The volume has the beautifully simple form V = ⁴⁄₃·π·a·b·c and for a = b = c it turns directly into the volume of a sphere. The surface area is far more complicated, though: for three different semi-axes it cannot be written with elementary functions and it leads to elliptic integrals. That is why an accurate approximation is used (Thomsen's formula, with an error of up to ~1.1 %). This calculator computes the surface area exactly whenever the solid is a sphere or a spheroid, and approximately in the remaining cases — and it always says which one it is.

Calculator

Length unit

Semi-axis acm

Semi-axis bcm

Semi-axis ccm

Volume of the ellipsoid V

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exact formula ⁴⁄₃·π·a·b·c

Surface area of the ellipsoid S

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Type of solid

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Axis ratio a : b : c

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Interactive graph

Drag the coloured points — they change the lengths of the semi-axes Re-center

Worked example

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Where do we find it

The Earth is an oblate spheroid — the radius at the equator is roughly 21 km larger than at the poles.
Geodesy uses a reference ellipsoid (for example WGS 84) as the model of the shape of the Earth for GPS.
A rugby or American football ball has the shape of a prolate spheroid.
Biology and medicine — the volume of cells, eggs or organs is often estimated with the ellipsoid formula.

Frequently asked questions

What is the formula for the volume of an ellipsoid?

V = ⁴⁄₃ · π · a · b · c, where a, b, c are the three semi-axes. If all of them are equal (a = b = c = r), the formula turns into the volume of a sphere ⁴⁄₃ · π · r³.

Why is there no simple formula for the surface area?

The surface area of a general ellipsoid with three different semi-axes cannot be expressed with elementary functions — it leads to elliptic integrals. That is why an approximation is used; Thomsen's formula has an error of up to roughly 1.1 %. For a sphere and for a spheroid exact formulas do exist and this calculator uses them.

What is a spheroid (an ellipsoid of revolution)?

It arises by rotating an ellipse about one of its axes, so two semi-axes are equal. If the third one is shorter, the solid is oblate (like the Earth), if it is longer, the solid is prolate (like a rugby ball).

How is an ellipsoid related to a sphere?

A sphere is a special case of an ellipsoid where a = b = c. An ellipsoid can therefore be pictured as a sphere stretched unevenly in three mutually perpendicular directions.

Practice problems

Solve it step by step — the next step unlocks only after a correct answer. You can type decimals with a dot or a comma.