Ellipsoid Volume Calculator

Calculate the volume and surface area of an ellipsoid from three semi-axes with step-by-step solutions.

Enter Ellipsoid Semi-axes

Results

Volume
--
cubic units
Surface Area (approx)--
Sphere Volume (r = a)--
Volume Ratio (ellipsoid/sphere)--
Cross-section area (xy)--
Cross-section area (xz)--
Cross-section area (yz)--
Is prolate/oblate/scalene?--

Step-by-Step Solution

V = (4/3) x pi x a x b x c

What Is an Ellipsoid?

An ellipsoid is a three-dimensional surface that is the generalization of an ellipse. It is described by the equation x2/a2 + y2/b2 + z2/c2 = 1, where a, b, and c are the three semi-axes along the x, y, and z directions respectively. When all three semi-axes are equal, the ellipsoid is a sphere.

Ellipsoid Formulas

Volume

The exact volume formula, derived from triple integration.

V = (4/3) x pi x a x b x c

Surface Area (Knud Thomsen)

An approximation accurate to about 1.061% maximum relative error.

S = 4*pi*((a^p*b^p + a^p*c^p + b^p*c^p)/3)^(1/p), p=1.6075

Sphere Comparison

A sphere is a special ellipsoid where a = b = c = r.

V_sphere = (4/3) x pi x r^3

Types of Ellipsoids

  • Sphere: a = b = c. All three semi-axes are equal.
  • Prolate spheroid: a = b < c (or two axes equal and smaller than the third). Shaped like a football or egg, elongated along one axis.
  • Oblate spheroid: a = b > c (or two axes equal and larger than the third). Shaped like a flattened sphere, like Earth.
  • Scalene ellipsoid (triaxial): All three semi-axes are different (a, b, c all unequal). The most general case.

Surface Area of an Ellipsoid

Unlike the volume, the surface area of a general ellipsoid has no simple closed-form expression. It requires elliptic integrals for exact computation. This calculator uses Knud Thomsen's approximation:

S = 4*pi * ((apbp + apcp + bpcp) / 3)1/p

where p = 1.6075. This approximation has a maximum relative error of about 1.061% for all ellipsoids, which is excellent for most practical purposes.

Derivation of the Volume Formula

The volume of an ellipsoid can be derived by transforming a unit sphere. The unit sphere x2 + y2 + z2 = 1 has volume (4/3)*pi. Scaling by a, b, and c along each axis multiplies the volume by abc, giving V = (4/3)*pi*abc. Alternatively, it can be derived using triple integration in Cartesian or spherical coordinates.

Real-World Applications

  • Earth and Planetary Science: Earth is approximately an oblate spheroid with equatorial radius 6378 km and polar radius 6357 km. Calculating its volume requires the ellipsoid formula.
  • Medicine: Tumors, organs (like the prostate), and other biological structures are often modeled as ellipsoids. Doctors use the formula V = (4/3)*pi*abc (or the simplified V = pi/6 * L*W*H with full axes) to estimate sizes from ultrasound or MRI measurements.
  • Engineering: Pressure vessels, fuel tanks, and aerodynamic bodies may have ellipsoidal shapes. Knowing the volume is essential for capacity calculations.
  • Geology: Sediment grains and mineral particles are often approximated as ellipsoids for size and volume calculations.
  • Astronomy: Many asteroids and minor planets are irregular but can be modeled as triaxial ellipsoids.

Examples

  1. Sphere (a = b = c = 5): V = (4/3)*pi*5*5*5 = (4/3)*pi*125 = 523.6 cubic units.
  2. Prolate (a = b = 4, c = 8): V = (4/3)*pi*4*4*8 = (4/3)*pi*128 = 536.17 cubic units.
  3. Earth (a = b = 6378 km, c = 6357 km): V = (4/3)*pi*6378*6378*6357 = 1.083 x 1012 km3.

Tips for Using This Calculator

  • The order of a, b, and c does not matter for volume calculation -- V = (4/3)*pi*abc is symmetric in all three axes.
  • For a sphere, set all three values equal. The result matches (4/3)*pi*r3.
  • The surface area is an approximation. For higher accuracy with spheroids, exact formulas involving elliptic integrals exist.
  • Cross-section areas are the areas of the ellipses formed by slicing through the center along each coordinate plane.