A stretched sphere
An ellipsoid is what you get when you stretch or squash a sphere along its axes. Eggs, rugby balls, the Earth, planets, many fruits and organs, and pressure tanks with domed ends all have shapes close to ellipsoids. Three semi-axes, a, b and c, measured from the center to the surface along three perpendicular directions, describe it completely.
Types of ellipsoids
- Sphere: a = b = c.
- Prolate spheroid: two short equal axes and one long axis, like a rugby ball or a grain of rice.
- Oblate spheroid: two long equal axes and one short axis, like the Earth or a lentil.
- Scalene (triaxial) ellipsoid: all three axes different, like many pebbles.
Volume
V = 4/3 × π × a × b × c
This formula is exact. For a sphere it becomes the familiar 4/3 × π × r³. If you measure full lengths instead of semi-axes, use V = π/6 × D₁ × D₂ × D₃, about 0.52 times the product of the three diameters. Radiologists use this ellipsoid formula to estimate the volume of the prostate, thyroid lobes, bladder and tumors on ultrasound or CT images.
Surface area
There is no simple exact formula for the surface area of a general ellipsoid. A very good approximation, published by Knud Thomsen, is:
S ≈ 4π × ((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ) ÷ 3)^(1/p), with p = 1.6075
Its error is at most about 1.06%, and much less for shapes close to a sphere.
Example
An ellipsoid with semi-axes of 5, 4 and 3 units:
- Volume = 4/3 × π × 5 × 4 × 3 = 251.3 cubic units
- Surface area ≈ 199.5 square units
- A sphere with the same volume would have a radius of (5 × 4 × 3)^(1/3) = 3.91
The Earth as an ellipsoid
The Earth spins, so it bulges at the equator. The WGS 84 reference ellipsoid used by GPS has an equatorial radius of 6,378.137 km and a polar radius of 6,356.752 km, a flattening of about 1 in 298. Using an ellipsoid instead of a sphere changes distance calculations by up to about 0.5%.
Tips
- Use the same unit for all three axes; volume comes out in that unit cubed and area in that unit squared.
- To estimate the capacity of a container, measure inside dimensions; 1 liter is 1,000 cm³.
- For a half-ellipsoid, such as a dome or a tank end, divide the volume by two.