About the Ellipsoid Calculator
An ellipsoid is a stretched or squashed sphere: every cross-section is an ellipse, and its size is fixed by three semi-axes a, b and c measured from the centre to the surface along three perpendicular directions. Rugby balls, eggs, watermelons, M&M-style candies, the shape of the Earth and many storage tanks and medical measurements (such as organ or tumour volumes) are modelled as ellipsoids.
Enter the three semi-axes and this ellipsoid calculator returns the volume, the exact surface area (computed with elliptic integrals, not a rough rule of thumb), the popular Knud Thomsen approximation for comparison, and the shape type — sphere, oblate spheroid, prolate spheroid or scalene (triaxial) ellipsoid. Pick a length unit to also see the capacity in litres and US gallons.
Semi-axes are half of the full widths. If you measured a full length, width and height (for example with calipers), halve each value first — or remember the ellipsoid volume equals π/6 × length × width × height.
With the default inputs, the volume is 251.3274. Change any value above to recalculate instantly.
How to use the ellipsoid calculator
- 1Measure the three semi-axes — half of the length, width and height through the centre.
- 2Enter a, b and c in any order (the calculator sorts them).
- 3Choose a length unit if you want litres and gallons.
- 4Read the volume, exact surface area and the shape type.
Formula and method
The volume of an ellipsoid is exact and simple: 4/3 · π · a · b · c, the sphere formula with r³ replaced by the product of the three semi-axes. Because a·b·c ÷ 8 = length × width × height of the bounding box, the same volume can be written as π/6 × L × W × H.
Surface area has no elementary formula. The calculator sorts the axes so that a ≥ b ≥ c and evaluates the exact Legendre expression S = 2πc² + (2πab ÷ sin φ)(E(φ, k) sin²φ + F(φ, k) cos²φ), with cos φ = c/a and k² = a²(b² − c²) ÷ (b²(a² − c²)), where F and E are incomplete elliptic integrals of the first and second kind (computed with Carlson’s symmetric forms). Knud Thomsen’s approximation with p = 1.6075 is shown alongside; its relative error never exceeds about 1.061%.
- a, b, c
- Semi-axes (half the full widths), sorted a ≥ b ≥ c for the area formula
- V
- Volume
- S
- Surface area
- F, E
- Incomplete elliptic integrals of the first and second kind
- p
- Thomsen exponent, about 1.6075
Worked examples
Triaxial ellipsoid a = 5, b = 4, c = 3
V = 4/3 × π × 5 × 4 × 3 = 80π ≈ 251.33 cubic units. The exact elliptic-integral surface area is about 199.46 square units, and Thomsen’s approximation gives 199.50 — within 0.03%. A sphere of radius ∛60 ≈ 3.915 has the same volume.
Rugby ball: 15 cm by 9.5 cm semi-axes (prolate spheroid)
A ball 30 cm long and 19 cm across has semi-axes 15, 9.5 and 9.5 cm. V = 4/3 × π × 15 × 9.5² ≈ 5,670 cm³, or about 5.67 litres, and the surface area is roughly 1,591 cm² — the amount of leather or rubber needed to cover it.
Earth as an oblate spheroid (WGS 84 semi-axes in km)
With an equatorial radius of 6,378.137 km and a polar radius of about 6,356.752 km, the Earth’s volume is about 1.0832 × 10¹² km³ and its surface area about 510.07 million km² — the familiar figures quoted for the WGS 84 reference ellipsoid.
Frequently asked questions
What is the formula for the volume of an ellipsoid?+
V = 4/3 × π × a × b × c, where a, b and c are the three semi-axes. For a = 5, b = 4 and c = 3 the volume is 80π ≈ 251.3 cubic units. If all three are equal it reduces to the sphere formula 4/3 πr³.
Is there an exact formula for ellipsoid surface area?+
Only in terms of elliptic integrals, which this calculator evaluates numerically. Spheroids (two equal axes) have closed forms using arcsine or logarithms, and Knud Thomsen’s formula gives a quick approximation within about 1.06%.
What is the difference between oblate and prolate spheroids?+
An oblate spheroid has two equal long axes and one shorter axis, like the Earth or a lentil. A prolate spheroid has two equal short axes and one longer axis, like a rugby ball or an egg.
How do I estimate volume from length, width and height?+
Use V = π/6 × L × W × H, which is the same as 4/3 π abc with each full measurement halved. Doctors use this “ellipsoid formula” (often written 0.52 × L × W × H) to estimate organ, cyst or prostate volumes from ultrasound.
Does the order of a, b and c matter?+
No. The volume depends only on the product a × b × c, and the calculator sorts the axes from largest to smallest before computing the surface area, so any order gives the same results.