Volume and Surface Area of an Ellipsoid

Finds the volume of an ellipsoid, a sphere with three different radii, as 4πabc ÷ 3. Its surface area cannot be written in elementary functions, so the Knud Thomsen approximation is used.

Stretch a sphere by a different factor along each of three axes and the result is an ellipsoid. A rugby ball is one, and so is the slightly flattened Earth.

The volume formula is the sphere's with $r^3$ replaced by $abc$ .

V = \dfrac{4\pi a b c}{3}

$V$ is the volume
$a$ , $b$ and $c$ are the three radii

Setting $a = b = c$ turns $abc$ back into $r^3$ and recovers the sphere. Volume scales directly with the product of the stretching factors, which is why the formula stays this simple.

The surface area cannot be written down

The surface area is another matter. No expression in elementary functions exists for it; elliptic integrals are required. It is the same obstacle that stops the perimeter of an ellipse being as easy as the circumference of a circle. Areas and volumes scale by multiplication, but lengths and surfaces bring in square roots that refuse to integrate cleanly.

An approximation is used instead. This calculator uses the formula of Knud Thomsen.

S \approx 4\pi \left(\dfrac{(ab)^{p} + (ac)^{p} + (bc)^{p}}{3}\right)^{\frac{1}{p}}, \quad p = 1.6075

The exponent 1.6075 was chosen to fit the true values. When all three radii are equal the bracket becomes the average of $(r^2)^p$ , giving exactly $4\pi r^2$ , so a sphere comes out with no error at all. Even in the worst case the error stays around 1.1%.

Example

The default input is an ellipsoid with radii 5, 4 and 3.

The volume is $4\pi \times 60 \div 3$ , about 251.3274. The approximation gives a surface area of about 199.5017.

The radius of a sphere of equal volume is also shown. It is the cube root of $abc$ , so the cube root of 60, about 3.9149. When only the amount of material matters, that figure makes it easy to compare against a sphere or another ellipsoid.

Points to watch

The surface area is approximate. Where an exact value is needed, an elliptic integral must be evaluated numerically.

The Earth's equatorial radius exceeds its polar radius by about 21 km, making it an ellipsoid with $a$ and $b$ equal and $c$ shorter. Shapes with two equal radii are called spheroids, and their surface area can be written in elementary functions after all. Only the fully general case resists.