Ellipsoid Calculator

Calculate volume, surface area, eccentricity, and other key properties of a 3D ellipsoid instantly.

Semi-Axis a: Semi-Axis b: Semi-Axis c:

Calculation Results

Enter ellipsoid dimensions to calculate all properties.

Ellipsoid Diagram

Ellipsoid: A 3D Oval Shape

An ellipsoid is a three-dimensional surface defined by three semi-axes (a, b, c). If all three semi-axes are equal, the shape becomes a perfect sphere. If two semi-axes are equal, the ellipsoid is called a spheroid.

Key Properties

Volume: V = (4/3)πabc
Surface Area: Approximated using Knud Thomsen’s formula
Equation: (x²/a²) + (y²/b²) + (z²/c²) = 1
Special Cases: Sphere (a = b = c), Oblate/Prolate Spheroids

How to Use the Ellipsoid Calculator

Enter the three semi-axes a, b, and c.
Click “Calculate” to get the results.
The calculator will instantly show Volume and Surface Area.
Click “Reset” to clear all inputs and start again.

Example Calculation

Suppose an ellipsoid has semi-axes:

a = 5 units
b = 4 units
c = 3 units

Volume:

V = (4/3) × π × 5 × 4 × 3 = 251.33 cubic units

Surface Area (approx.):

S ≈ 4π × ((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ) / 3)^(1/p), where p ≈ 1.6075

After calculation → S ≈ 237.89 square units

Derivation of Ellipsoid Formulas

1. Equation of an Ellipsoid

An ellipsoid centered at the origin with semi-axes a, b, and c is given by:

(x² / a²) + (y² / b²) + (z² / c²) = 1

2. Volume of an Ellipsoid

The ellipsoid can be thought of as a stretched sphere. If a sphere of radius r has volume V = (4/3)πr³, then scaling along x, y, z axes gives:

V = (4/3)πabc

Proof (integral form): V = ∬∬ dV inside ellipsoid = (4/3)πabc

3. Surface Area of an Ellipsoid

Unlike volume, there is no simple closed form for ellipsoid surface area. The exact formula involves elliptic integrals, but an excellent approximation is:

S ≈ 4π × ((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ) / 3)^1/p, where p ≈ 1.6075 (Knud Thomsen's formula).

Special Cases of an Ellipsoid

1. Sphere (a = b = c = r)

When all three semi-axes are equal, the ellipsoid becomes a sphere of radius r.

Volume: V = (4/3)πr³
Surface Area: S = 4πr²

2. Prolate Spheroid (a = b < c)

An ellipsoid stretched along the z-axis (like a rugby ball).

Equation: (x² / a²) + (y² / a²) + (z² / c²) = 1
Volume: V = (4/3)πa²c
Surface Area: S = 2πa² + (2πac / e)·sin⁻¹(e), where e = √(1 − (a² / c²)) is the eccentricity.

3. Oblate Spheroid (a = b > c)

An ellipsoid flattened along the z-axis (like the Earth).

Equation: (x² / a²) + (y² / a²) + (z² / c²) = 1
Volume: V = (4/3)πa²c
Surface Area: S = 2πa² + (πc² / e)·ln((1+e)/(1−e)), where e = √(1 − (c² / a²)).

🌍 Real-World Example

The Earth is an oblate spheroid, with its equatorial radius larger than its polar radius.

Frequently Asked Questions about Ellipsoids

1. Is an ellipsoid always symmetrical?

Yes, ellipsoids are symmetrical about all three coordinate planes.

2. What is the difference between a sphere and an ellipsoid?

A sphere is a special case of an ellipsoid where a = b = c.

3. Can the surface area be calculated exactly?

No simple formula exists. Approximations like Knud Thomsen’s formula are used.

4. Where are ellipsoids used?

In physics, astronomy, and engineering (e.g., Earth is an oblate spheroid).