Ellipsoid Volume Calculator

Compute the volume of any ellipsoid with semi-axes a, b, c. Real‑time calculation, multiple unit support, and 3D axis visualization. Ideal for students, engineers, and geoscientists.

Ellipsoid volume formula: V = ⁴⁄₃ · π · a · b · c

where a, b, c are the semi‑axis lengths (radii) along x, y, z directions.

Ellipsoid (5,3,2) Earth (km) WGS84 Sphere (r=10) Prolate spheroid Oblate spheroid Rugby ball
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Understanding the Ellipsoid

An ellipsoid is a three‑dimensional surface whose plane sections are ellipses. It is defined by three perpendicular semi‑axes: a (x‑direction), b (y‑direction), and c (z‑direction). When all three are equal, it becomes a sphere.

Volume formula derivation: The volume of an ellipsoid is exactly ⁴⁄₃πabc. This can be derived by integrating or by scaling a sphere: a sphere of radius 1 has volume ⁴⁄₃π; scaling the x, y, z axes by a, b, c multiplies volume by a·b·c.

Special Cases

  • Sphere: a = b = c → V = ⁴⁄₃πa³
  • Oblate spheroid: a = b > c (Earth‑like, flattened at poles)
  • Prolate spheroid: a = b < c (cigar‑shaped)
  • Scalene ellipsoid: all three different

Real‑World Applications

1
Geodesy: Earth is approximated as an oblate spheroid (a≈b≈6378 km, c≈6356 km).
2
Medicine: Tumor or organ volume estimation from MRI scans.
3
Engineering: Pressure vessel design, storage tanks.
4
Physics: Moment of inertia calculations.

Did you know? The surface area of an ellipsoid does not have a simple closed‑form formula; it requires elliptic integrals. This calculator focuses on exact volume.

Frequently Asked Questions

You can input axes in mm, cm, m, km, inches, or feet. Volume is shown in the selected output unit (including liters and gallons). Conversions are done automatically using 1 L = 1000 cm³, 1 gal (US) = 3785.41 cm³, etc.

Surface area of an ellipsoid is more complex. For spheroids (a=b or b=c) there are formulas, but for triaxial ellipsoids we recommend numerical integration. This tool focuses on exact volume.

Semi‑axes must be positive. Negative values are treated as positive (absolute value) and a warning is shown to ensure correct interpretation.