Ellipsoid Volume Calculator

An ellipsoid is a 3D shape defined by three semi-axes: a sphere stretched, or squashed, along three perpendicular directions.

V = (4 / 3) × π × a × b × c

Where a, b, c are the three semi-axes (half-lengths along the three perpendicular axes).

When a = b = c = r, the ellipsoid is a sphere with V = (4/3)πr³. ✓

Worked example: a rugby ball A regulation rugby ball is approximately a prolate spheroid (a = b, c ≠ a). For a Size 5 (adult) rugby ball: length 290 mm (c = 145 mm), girth 770 mm circumference → diameter 245 mm (a = b = 122.5 mm). V = (4/3) × π × 122.5 × 122.5 × 145 ≈ 9,114,000 mm³ ≈ 9.11 L.

That’s the interior air volume of a properly inflated rugby ball. Compare to a soccer ball (sphere with diameter 220 mm): V = (4/3)π × 110³ ≈ 5.58 L. The rugby ball is about 60% bigger.

Worked example: a chicken egg A large chicken egg is approximately ellipsoidal: a = b ≈ 21.5 mm, c ≈ 28.5 mm. V = (4/3) × π × 21.5 × 21.5 × 28.5 ≈ 55,200 mm³ ≈ 55 mL.

This matches the typical reported “large egg” volume of 50-60 mL, which is why one large egg ≈ 1/4 cup of egg white + yolk in cooking math.

Worked example: Earth as an oblate spheroid Earth is slightly flattened at the poles due to rotation: equatorial radius a = b = 6,378.137 km, polar radius c = 6,356.752 km. V = (4/3) × π × 6,378² × 6,357 ≈ 1.083 × 10¹² km³.

Now compare that against a perfect sphere of the quoted mean radius, 6,371 km: (4/3)π × 6,371³ ≈ 1.083207 × 10¹² km³. The two agree to about three parts in ten million.

That is not luck. The mean radius is defined as (2a + c)/3 = 6,371.0 km precisely so that the sphere and the ellipsoid come out to the same volume, which is why you can use 6,371 km for volume work and never think about Earth’s flattening again.

Compare instead against a sphere of the equatorial radius, 6,378 km, and the gap opens to 0.34%. That is the difference people usually have in mind, and it is a difference in which radius you picked, not in how flattened the planet is.

Where ellipsoids appear in real measurements:

• Eggs and ovoid foods. Chicken eggs, quail eggs, kiwi fruit. Approximately ellipsoidal.
• Rugby and American football balls. Prolate spheroids (longer than wide).
• Planet shapes. Earth, Mars and Saturn are all oblate spheroids.
• Coffee beans, almonds, peanuts. Roughly ellipsoidal seeds.
• Pharmaceutical caplets. Smooth-ended tablets shaped between a sphere and a capsule.
• Submarine hull shapes. Prolate spheroid for low drag.
• Eyeballs. Roughly spherical, but slightly ellipsoidal (radius along eye axis differs from radius perpendicular).

Three special cases:

1. Sphere (a = b = c = r): V = (4/3)πr³.
2. Oblate spheroid (a = b > c, flattened at poles): planets, lentils, coins.
3. Prolate spheroid (a = b < c, stretched along one axis): rugby balls, footballs, eggs.

For ellipsoids where all three semi-axes differ, called scalene or triaxial, the formula still applies but the shape is harder to picture. Asteroids and irregular planetoids are often triaxial.

Ellipsoid vs. sphere of equivalent volume:

For our rugby ball: V = 9.11 L. An equivalent-volume sphere has r = (3V / 4π)^(1/3) = (3 × 9110 / 4π)^(1/3) ≈ 130 mm. So the rugby ball has roughly the same volume as a 260 mm diameter ball.

Sanity check:

• Any semi-axis = 0: V = 0. ✓
• a = b = c = r (sphere): V = (4/3)πr³. ✓
• For a = 1, b = 2, c = 3: V = (4/3)π × 6 = 8π ≈ 25.13. (Unit-style scalene ellipsoid.)