Ellipsoid Volume Calculator

An ellipsoid is a 3D shape defined by three semi-axes — like a sphere stretched (or compressed) 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 — 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 — 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 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.

The difference from a perfect sphere with r = 6,371 km (average radius): about 0.3% — small but measurable.

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, Saturn — 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 (a ≠ b ≠ c) — also called “scalene” or “triaxial” ellipsoids — the formula still applies but the shape is harder to visualize. 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.)