Ellipsoid Volume Calculator

Enter the three semi-axes of an ellipsoid to get its volume (V = 4/3 × π × a × b × c), an approximate surface area, and the radius of an equivalent-volume sphere.

Quick Facts

Volume formula
V = (4/3)πabc
a, b, c are the three semi-axes measured from the center to the surface.
Sphere special case
a = b = c = r → V = (4/3)πr³
Equal semi-axes reduce the ellipsoid formula to the sphere formula.
Surface area (approx.)
Thomsen's formula, p ≈ 1.6075
No simple exact formula exists; this approximation is accurate to within ~1.1%.

Your Results

Calculated
Volume
-
V = (4/3)πabc, in cubic units
Approx. Surface Area
-
Thomsen's approximation, in square units
Equivalent-Volume Sphere Radius
-
r = ∛(abc), same volume as a sphere
Volume in Liters
-
Useful for tank or container capacity

Ready

Enter the three semi-axes and unit, then press Calculate.

Formula and Method for Ellipsoid Volume

An ellipsoid is a three-dimensional shape where every planar cross-section is an ellipse (or a circle). It is defined by three semi-axes — a, b, and c — the distances from the center to the surface along three mutually perpendicular directions. Its volume is V = (4/3)πabc, a direct generalization of the sphere formula V = (4/3)πr³, which is the special case where a = b = c = r.

How the calculation works

Enter the three semi-axis lengths and the unit they are measured in. The calculator multiplies a × b × c, multiplies that product by 4/3 and π, and reports the volume in cubic units of whatever unit you selected. It also estimates the surface area using Thomsen's formula, S ≈ 4π[((apbp + apcp + bpcp)/3)]1/p with p ≈ 1.6075 — there is no simple exact closed-form formula for an ellipsoid's surface area, but this approximation stays within about 1.1% of the true value for any combination of axis lengths. Finally, it reports the radius of a sphere with the same volume, r = ∛(abc), and converts the volume to liters using the standard 1 m³ = 1000 L relationship.

Common mistakes

  • Semi-axis vs. full axis: a, b, and c are half-lengths (center to surface), not the full length across the shape. If you measured the full width, height, or depth, divide by 2 before entering it.
  • Units: volume comes out in cubic units (cm³, m³, etc.) and surface area in square units. Keep all three semi-axes in the same unit before calculating.
  • Assuming a sphere formula works: using a single radius when the three axes actually differ under- or over-estimates the volume — always measure all three directions for an elongated or flattened object.

Real-world applications

  • Estimating the volume of eggs, seeds, pills, and other roughly ellipsoidal biological or industrial objects
  • Tank, vessel, and storage capacity planning for elliptical or dome-shaped containers
  • Geodesy and planetary science, where Earth is modeled as an oblate spheroid (a = b > c)
  • Medical imaging, where tumor or organ volumes are frequently approximated from three measured diameters

Frequently Asked Questions

What is the formula for the volume of an ellipsoid?
The volume of an ellipsoid is V = (4/3)πabc, where a, b, and c are the lengths of the three semi-axes (the distances from the center to the surface along each of the three perpendicular axes). For example, semi-axes of 5, 3, and 2 cm give V = (4/3)×π×5×3×2 ≈ 125.66 cm³.
How is an ellipsoid different from a sphere?
A sphere is a special case of an ellipsoid where all three semi-axes are equal (a = b = c = r), which reduces the formula to the familiar V = (4/3)πr³. When the three semi-axes differ, the shape is stretched or flattened unevenly along each axis, giving an egg-like or flattened form instead of a perfectly round one.
What is the formula for the surface area of an ellipsoid?
Unlike volume, there is no simple exact formula for an ellipsoid's surface area in general. This calculator uses Thomsen's approximation, S ≈ 4π[((apbp + apcp + bpcp)/3)]1/p with p ≈ 1.6075, which stays within about 1.1% of the true value for any combination of semi-axes.
What if two of the semi-axes are equal (a spheroid)?
When two semi-axes are equal (a = b ≠ c), the ellipsoid is a spheroid and the volume formula simplifies to V = (4/3)πa²c. If c is longer than a, it is a prolate spheroid (football-shaped); if c is shorter, it is an oblate spheroid (disc-shaped, like Earth's approximate shape).

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