What is Knud Thomsen's approximation?

Knud Thomsen's formula approximates the surface area of a general ellipsoid as S ≈ 4π × ((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ) / 3)^(1/p), where p ≈ 1.6075. There is no exact closed-form formula for a general ellipsoid, so this approximation — accurate to within about 1.061% — is widely used in practice.

Is there an exact formula for an ellipsoid's surface area?

Only for special cases. A sphere (a = b = c) uses S = 4πr². A prolate spheroid (a = b ≠ c) and an oblate spheroid (a ≠ b = c) each have exact formulas involving elliptic integrals, but the fully general case requires numerical integration or an approximation like Knud Thomsen's.

How accurate is this calculator?

Knud Thomsen's approximation has a maximum relative error of about 1.061% across all ellipsoid shapes. For most practical applications — engineering, education, estimation — this accuracy is more than sufficient.

A semi-axis is half the length of a principal diameter of the ellipsoid. For a sphere of radius r, all three semi-axes are equal to r. For an egg-shaped prolate spheroid, the longest semi-axis (the pole-to-center distance) is greater than the other two.

What happens when all three axes are equal?

When a = b = c = r, the ellipsoid is a sphere and the formula returns S = 4πr², which is the exact sphere surface area formula. You can verify this with this calculator.

Do the axes need to be in a specific order?

No. The formula is symmetric with respect to permutations of a, b, and c — entering them in any order gives the same result.

Surface Area of an Ellipsoid Calculator

Name: Surface Area of an Ellipsoid Calculator
Availability: InStock
Author: Nham Vu

Enter the three semi-axes a, b, and c of an ellipsoid to instantly compute its surface area using Knud Thomsen's approximation.

Ellipsoid Semi-Axes

Semi-axis a (x-direction)

Semi-axis b (y-direction)

Semi-axis c (z-direction)

Unit

Surface Area

—

Enter values and click Calculate

Ellipsoid diagram

a, b, c = semi-axes along x, y, z directions

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Summary

Enter the three semi-axes a, b, and c of an ellipsoid to instantly compute its surface area using Knud Thomsen's approximation.

How it works

Enter semi-axis a — the radius along the x-axis.
Enter semi-axis b — the radius along the y-axis.
Enter semi-axis c — the radius along the z-axis.
Choose a length unit (cm, m, mm, in, or ft).
Click "Calculate" — the surface area appears with the step-by-step working.
Use the copy button to copy the result to your clipboard.

Use cases

Estimating the surface area of eggs, planets, or sporting balls.
Engineering calculations for tank and vessel designs with ellipsoidal heads.
Medical imaging — approximating organ surface areas from measured radii.
Packaging design for ellipsoidal products.
Physics and astronomy homework involving oblate or prolate spheroids.
Verifying hand calculations for three-dimensional geometry problems.
Teaching the difference between exact and approximate surface area formulas.
Manufacturing cost estimation based on material surface coverage.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu