Ellipsoid Surface Area Calculator

Calculate the surface area of an ellipsoid by entering the three semi-axes. An ellipsoid is a stretched or compressed sphere with three different radii.

🥚 Ellipsoid Surface Area

Metric (cm, m) Imperial (in, ft)

Semi-axis A (cm)

Semi-axis B (cm)

Semi-axis C (cm)

Show calculation steps Show volume calculation Show approximation method

Formula

Knud Thomsen's Approximation:

SA ≈ 4π × ((a^p × b^p + a^p × c^p + b^p × c^p) / 3)^(1/p)

where p ≈ 1.6075

Where a, b, c are the three semi-axes of the ellipsoid

Ellipsoid Surface Area Formulas

Knud Thomsen's Approximation

SA ≈ 4π((a^p·b^p + a^p·c^p + b^p·c^p)/3)^(1/p)

Highly accurate approximation with p ≈ 1.6075

Sphere (Special Case)

SA = 4πr² (when a = b = c = r)

When all semi-axes are equal, it becomes a sphere

Volume

V = (4/3)πabc

Volume of an ellipsoid

Prolate Spheroid

Special case when a = b ≠ c

Football or rugby ball shape

Ellipsoid Surface Area Examples

Example 1: Prolate Spheroid

Given: a = 3 cm, b = 3 cm, c = 5 cm (football shape)

Using Thomsen's approximation with p = 1.6075

Result: SA ≈ 4π × ((3^1.6075 × 3^1.6075 + 3^1.6075 × 5^1.6075 + 3^1.6075 × 5^1.6075) / 3)^(1/1.6075)

Surface Area: ≈ 158.73 cm²

Example 2: Oblate Spheroid

Given: a = 4 cm, b = 4 cm, c = 2 cm (flattened sphere)

Using Thomsen's approximation

Surface Area: ≈ 140.25 cm²

Example 3: General Ellipsoid

Given: a = 2 cm, b = 3 cm, c = 4 cm

Using Thomsen's approximation

Surface Area: ≈ 111.84 cm²

Ellipsoid Surface Area FAQ

What is an ellipsoid?

An ellipsoid is a 3D shape that looks like a stretched or compressed sphere. It has three semi-axes (a, b, c) that can be different lengths, creating various oval shapes.

Why is the surface area formula an approximation?

Unlike simple shapes, ellipsoids don't have a closed-form exact formula for surface area. Knud Thomsen's approximation is extremely accurate (error < 1.061%) for all ellipsoids.

What are prolate and oblate spheroids?

A prolate spheroid has two equal shorter axes (football shape: a = b < c). An oblate spheroid has two equal longer axes (flattened sphere: a = b > c).

How accurate is Thomsen's approximation?

Knud Thomsen's formula with p ≈ 1.6075 has a maximum relative error of about 1.061%, making it highly accurate for practical applications.