An ellipsoid is a stretched or squashed sphere: every cross-section through its center is an ellipse. Eggs, watermelons, rugby balls, planets and even the shape used to model the Earth for GPS are ellipsoids or close to them. The volume formula is simple, but the surface area of a general ellipsoid has no elementary formula. This calculator computes the volume and an essentially exact surface area using elliptic integrals, names the type of ellipsoid, and compares with a popular approximation.
How to use the ellipsoid calculator
- Choose whether your measurements are semi-axes (center to surface) or full axes (overall length, width and height).
- Enter a, b and c in any order and pick the unit.
- Read the volume on the tape, with the surface area, the shape type, the equal-volume sphere radius, Thomsen’s approximation and the capacity. The diagram shows the three axes.
Ellipsoid formulas
The surface area depends on the shape:
| Shape | Axes | Surface area |
|---|---|---|
| Sphere | a = b = c | 4πa2 |
| Oblate spheroid | a = b > c | 2πa2[1 + ((1 − e2)/e) artanh e], e2 = 1 − c2/a2 |
| Prolate spheroid | a > b = c | 2πb2[1 + (a/(be)) arcsin e], e2 = 1 − b2/a2 |
| Triaxial | all different | elliptic-integral formula below |
For a triaxial ellipsoid with a ≥ b ≥ c, let cos φ = c/a and k2 = a2(b2 − c2)/(b2(a2 − c2)). Then
where F and E are incomplete elliptic integrals of the first and second kind. The calculator evaluates them with Carlson’s symmetric forms, accurate to many more digits than you will ever need.
Worked example
An ellipsoid has semi-axes 5, 4 and 3 cm (overall 10 × 8 × 6 cm).
Volume: (4/3)π × 5 × 4 × 3 = 80π ≈ 251.33 cm³, about a quarter of a liter.
Surface area: all three axes differ, so the elliptic-integral formula applies: S ≈ 199.46 cm².
Thomsen's approximation: ≈ 199.50 cm², only 0.02% high for this fairly round shape.
Equal-volume sphere: radius ∛(5 × 4 × 3) = ∛60 ≈ 3.91 cm.
A spheroid check: with a = b = 2 and c = 1 (an oblate spheroid), e = √0.75 ≈ 0.866 and the closed formula gives about 34.69 square units, matching the general method exactly.
Real-world ellipsoids
- Earth is modeled as an oblate spheroid with an equatorial radius of 6,378.137 km and a polar radius of about 6,356.752 km (the WGS 84 reference ellipsoid used by GPS) — a flattening of only about 1 part in 298.
- Eggs and fruit are often approximated as prolate spheroids to estimate volume and surface for packaging or cooking.
- Tanks and domes with elliptical heads use half-ellipsoids; divide the volume by two for one head.
- Medicine uses the ellipsoid volume formula to estimate organ and tumor sizes from three measured diameters, often written as π/6 × length × width × height.
That last version is the same formula expressed with full diameters: (4/3)π(L/2)(W/2)(H/2) = (π/6)LWH.
Related tools
When all three axes are equal, the sphere calculator is simpler. Any slice through the center is an ellipse, handled by the ellipse calculator; a slice off the top of a sphere is a spherical cap. For other solids, see the volume calculator and the surface area calculator.
Frequently asked questions
What is the volume formula for an ellipsoid?
V = (4/3)πabc, where a, b and c are the three semi-axes (half the full widths). With a = 5, b = 4 and c = 3, the volume is 80π ≈ 251.33 cubic units.
Is there a simple formula for the surface area?
Only for spheres and spheroids. A general ellipsoid's surface area needs elliptic integrals, which the calculator evaluates numerically. Knud Thomsen's approximation 4π[((ab)^p + (ac)^p + (bc)^p)/3]^(1/p) with p ≈ 1.6075 is within about 1.06%.
What is the difference between oblate and prolate?
An oblate spheroid has two equal long axes and one short one, like a lentil or Earth, which is slightly flattened at the poles. A prolate spheroid has one long axis and two equal short ones, like a rugby ball.
Should I enter semi-axes or full axes?
Either. Choose 'Full axes (diameters)' if you measured overall length, width and height, and the calculator halves them for you.