Last updated: 2026-06-04

Definition

An ellipsoid is a closed, smooth surface in three dimensions whose cross-sections are all ellipses. It is defined by three semi-axes — half-lengths along the three coordinate directions:

• a — the longest semi-axis (along the x-axis)
• b — the middle semi-axis (along the y-axis)
• c — the shortest semi-axis (along the z-axis, depth)

Every point on the surface satisfies x²/a² + y²/b² + z²/c² = 1. When all three semi-axes are equal (a = b = c = r), the ellipsoid becomes a sphere.

Common real-world examples include the Earth (an oblate spheroid, a ≈ b > c), a rugby ball (a prolate spheroid, a > b ≈ c), a cell nucleus, a planet, and an engineering pressure vessel.

Formulas

Volume derivation

The volume formula V = (4/3)πabc can be derived by integrating elliptical cross-sections. At height z, the horizontal cross-section is an ellipse with semi-axes a·√(1 − z²/c²) and b·√(1 − z²/c²), so its area is πab(1 − z²/c²). Integrating from z = −c to z = c:

V = ∫₋ᶜᶜ πab(1 − z²/c²) dz = πab[z − z³/(3c²)]₋ᶜᶜ = πab · (4c/3) = (4/3)πabc

This is a direct generalization of the sphere formula: replace each r with the corresponding semi-axis.

Why the surface area has no exact closed form

For a sphere, every latitude circle has the same curvature relationship, and the surface integral simplifies to 4πr². For an ellipsoid the curvature varies continuously from point to point, and the surface integral reduces to incomplete elliptic integrals of the first and second kind — transcendental functions with no elementary closed form.

For the special case of a spheroid (a = b ≠ c), the integrals reduce to elementary expressions:

• Oblate spheroid (a = b > c): A = 2πa²(1 + (c²/(a²e))·arctanh e), where e = √(1 − c²/a²)
• Prolate spheroid (a > b = c): A = 2πc²(1 + (a/(ce))·arcsin e), where e = √(1 − c²/a²)

For a triaxial ellipsoid (a ≠ b ≠ c), a practical approximation is required.

Knud Thomsen approximation

The most accurate simple approximation for the surface area of a triaxial ellipsoid is the Knud Thomsen formula (2004):

A ≈ 4π · [(aᵖbᵖ + aᵖcᵖ + bᵖcᵖ) / 3]^(1/p), where p = 1.6075

The exponent p = 1.6075 was chosen empirically to minimize the worst-case relative error. For any combination of positive semi-axes the maximum error is below 1.061 %, and for a sphere (a = b = c) the formula is exact — it reduces algebraically to 4πa².

Special cases

Worked example

An ellipsoidal aquarium tank has semi-axes a = 40 cm, b = 25 cm, c = 20 cm.

Volume: V = (4/3) × π × 0.40 × 0.25 × 0.20 = (4/3) × π × 0.020 ≈ 0.08378 m³ ≈ 83.8 liters

Surface area (Knud Thomsen, p = 1.6075):

• 0.40^1.6075 ≈ 0.2293; 0.25^1.6075 ≈ 0.1077; 0.20^1.6075 ≈ 0.07523
• aᵖbᵖ = 0.02469; aᵖcᵖ = 0.01725; bᵖcᵖ = 0.008102
• Sum/3 = 0.01668; ^(1/1.6075) ≈ 0.07835

A ≈ 4π × 0.07835 ≈ 0.985 m² (≈ 9 846 cm²)

This figure tells the manufacturer how much acrylic sheet is needed to fabricate the tank wall.

Volume scales with the product of all three axes

Because V = (4/3)πabc, doubling any single semi-axis doubles the volume — unlike a sphere where doubling r multiplies V by 8. Doubling all three semi-axes simultaneously does multiply the volume by 8, consistent with a uniform scale factor of 2.