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Ellipsoid Calculator

Calculate ellipsoid volume and surface area. Solve for any semi-axis using the V = 4/3 pi abc formula with Thomsen surface area approximation.

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What is the Ellipsoid Calculator?

The Ellipsoid Calculator is a free online geometry tool for computing ellipsoid volume and surface area. It uses the formula V = (4/3)πabc for volume and Knud Thomsen's approximation for surface area. You can also solve for any semi-axis when volume and the other two axes are known.

This calculator is valuable for earth scientists, medical professionals, optical engineers, and students working with three-dimensional elliptical shapes.

How to Use the Ellipsoid Calculator

Select what you want to solve for from the dropdown. Enter the three semi-axes (half of the full diameters). The calculator instantly computes volume and surface area with detailed step-by-step calculations.

Understanding Ellipsoid Formulas

Volume: V = (4/3)πabc, generalizing the sphere formula V = (4/3)πr³.

Surface Area (Thomsen): S ≈ 4π · [((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)]^(1/p) with p = 1.6075. Worst-case error under 1.061%.

Applications of Ellipsoid Geometry

  • Earth science: Earth modeled as an oblate spheroid (WGS84 reference ellipsoid)
  • Medical imaging: organ volume estimation from radiology scans
  • Optics and astronomy: telescope mirrors and planetary shapes
  • Manufacturing: pressure vessels and container sizing
  • Sports: ball and equipment shape analysis

Frequently Asked Questions

How do you calculate the volume of an ellipsoid?

Multiply (4/3)π by all three semi-axes: V = (4/3)πabc. For a = 3 m, b = 2 m, c = 1 m, V = (4/3) × π × 6 = 8π ≈ 25.13 m³.

What is the formula for ellipsoid surface area?

There is no exact closed-form formula. Knud Thomsen's approximation S ≈ 4π · [((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)]^(1/p) with p = 1.6075 has worst-case error under 1.061% and recovers the exact sphere value 4πr² when a = b = c.

What is the difference between ellipsoid and spheroid?

A spheroid has two equal semi-axes (oblate: a = b > c, like Earth; prolate: a > b = c, like a rugby ball). A general ellipsoid has all three semi-axes different (a ≠ b ≠ c), called a triaxial ellipsoid.

Is Earth an ellipsoid?

Approximately yes. Earth is modeled as an oblate spheroid with equatorial radius ≈ 6378.137 km and polar radius ≈ 6356.752 km (flattening of about 1/298.257). The GPS reference model is WGS84.

How accurate is the Thomsen surface approximation?

With exponent p = 1.6075, worst-case relative error is about 1.061% across all aspect ratios. For typical engineering aspect ratios (under 10:1) the error is often under 0.1%.

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