# Surface Area and Volume of an Ellipsoid

An ellipsoid is a sphere-like solid shape that is the 3-d analogue of an ellipse. Whereas a sphere has a single radius, an ellipsoid has three semi-axes: half-height, half-length, and half-width. If A, B, and C represent each semi-axis, then the volume and surface area of an ellipsoid are functions of A, B, and C.

## Volume of an Ellipsoid

The volume of an ellipsoid is easy to learn and remember because it is so similar to that of a sphere. Recall that a sphere's volume is (4*pi/3)r^3. If you replace each of the three r's with A, B, and C, you get the formula for ellipsoid volume:

**V = (4*pi/3)ABC**

** **

## Ellipsoid Surface Area Equations

The formula for the volume of an ellipsoid is much more complicated than the analogous formula for a sphere. A sphere's surfaces area is a simple quadratic function in r, S(r) = 4*pi*r^2, however no such quadratic function in A, B, and C exists for computing the surface area of an ellipsoid.

Complicating the matter is that the surface area formula is different depending on the relative sizes of A, B, and C. If all three axis lengths are distinct there is actually no closed form expression involving elementary functions, but you can express the surface area in terms of elliptic integrals.

If the ellipsoid is oblate or prolate, cases in which two axis lengths are equal, then there there are closed-form expressions. These types of ellipsoids are called spheroids because they can be obtained as solids of revolution by spinning an ellipse about one of its axes.

## Oblate Ellipsoid (Oblate Spheroid)

An oblate ellipsoid is one for which the semi-axis lengths are A, A, and B, where B is less than A. This type is spheroid is flattened like an M&M candy. The formula for the surface area of an oblate spheroid is

**S = 2*pi*[1 + (1-m^2)arctanh(m)/m]*A^2**

where arctanh is the inverse hyperbolic tangent function given by

arctanh(x) = 0.5*Ln[(x+1)/(1-x)]

and m = sqrt(1 - (B/A)^2).

## Prolate Ellipsoid (Prolate Spheroid)

A prolate spheroid is one for which the axis lengths are A, B, and B, where B is less than A. This type of spheroid is elongated like a rugby ball. The formula for the surface area of a prolate ellipsoid is

**S = 2*pi*[1 + A*arcsin(m)/(mB)]*B^2**

where arcsin is the inverse sine function and m = sqrt(1 - (B/A)^2).

## General Ellipsoid

If all three axis lengths A, B, and C are unequal, with A greater than B and B greater than C, then the formula for the surface area is

In the formula, E(φ, k) is the incomplete elliptic integral of the second kind and F(φ, k) is the incomplete elliptic integral of the first kind. Neither can be evaluated in terms of simpler known functions, but they can be computed with convergent algorithms.

You can approximate the surface area of an ellipsoid with the equation

## More Geometry Tutorials

Related articles in finding the areas and volumes of different shapes

- How to Compute the Surface Area of Revolution About the Y-Axis
- How to Compute the Perimeter of an Ellipse: Integral and Approximation Formulas
- How to Find the Area, Perimeter, and Diagonal of a Rectangle
- How to Find the Area, Perimeter, and Diagonal of a Rectangle
- How to Find the Area of a Triangle with 3 Coordinate Points

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## Comments 5 comments

For the general ellipsoid case, with sem-axis A greateer than B greater than C, is it true to say an upper bound on the surface area is S(r) = 4*pi*A^2 and a lower bound is S(r) = 2*pi*BC? Is it more restrictive than that?

I meant the two. Consider, if that third axis is significantly smaller than the other two, then what you are looking at is the area of the topside and bottom side of an ellipse, or twice the area of one - pi*BC (or would it be pi*AB?). That would make a good test question, would it not? Prove or disprove the following conjecture ...

5