What is the formula for the volume of an ellipsoid?

The volume of an ellipsoid with semi-axes a, b, and c is given by V = 4/3 · π · a · b · c. If you use diameters D1, D2, and D3 (full axis lengths), the formula can be written as V = π/6 · D1 · D2 · D3, which is algebraically equivalent to the semi-axis formula.

What units does the ellipsoid volume calculator use?

You can enter the axes in millimeters, centimeters, meters, inches, or feet. The calculator computes the volume in the corresponding cubic unit (for example, cm³) and also converts it to cubic meters and liters where appropriate. All three axes must be in the same length unit for the formulas to be valid.

Can this calculator solve for an unknown axis of an ellipsoid?

Yes. If you know the volume of the ellipsoid and two of the semi-axes, you can use the inverse mode to solve for the third semi-axis. The tool rearranges the formula V = 4/3 · π · a · b · c to compute the missing dimension.

How is the ellipsoid volume used in practice?

Ellipsoid volume formulas are widely used in geometry, calculus, and engineering to approximate the volume of smoothly rounded objects. In radiology and medicine, a variant of the same formula is used to estimate the volume of organs or lesions from three orthogonal diameters. In physics and materials science, ellipsoids are used to approximate particle shapes and derive effective properties.

What is the difference between a sphere and an ellipsoid?

A sphere is a special case of an ellipsoid where all three semi-axes are equal: a = b = c = r. In that case, the ellipsoid volume formula reduces to V = 4/3 · π · r³, the familiar volume of a sphere. In general ellipsoids, the three axes can have different lengths, leading to elongated or flattened shapes.

Does this calculator handle very small or very large ellipsoids?

Yes. The calculator uses standard double-precision floating point arithmetic, which comfortably covers most practical dimensions from microscopic scales to large engineering structures. Very extreme values are rounded for display; for critical design problems, you should still verify results with dedicated engineering or numerical software.

Ellipsoid Volume Calculator

V = 4/3 · π · a · b · c

Compute the volume of an ellipsoid from its three axes, either as semi-axes (a, b, c) or full diameters. You can also invert the formula to solve for one semi-axis when the volume and the other two semi-axes are known.

Interactive ellipsoid volume calculator

Calculation mode

Use semi-axes if you have radii along each principal direction. Use diameters if you have full lengths (for example radiology long × width × height measurements).

Length unit

All axes must be in the same length unit. The main result is given in that unit cubed and converted to m³ and liters.

Semi-axes a, b, c (radii)

a (semi-axis)

b (semi-axis)

c (semi-axis)

These are half the lengths of the principal axes of the ellipsoid. The calculator applies \(V = \frac{4}{3}\pi a b c\).

This tool uses \(V = \frac{4}{3}\pi a b c\) with IEEE double-precision arithmetic. For high-stakes engineering or clinical decisions, always validate results with your organisation’s approved software or nomograms.

Ellipsoid volume formula

An ellipsoid is the three-dimensional analogue of an ellipse. It is defined by three semi-axes \(a\), \(b\), and \(c\) along three orthogonal directions. The volume is:

\[ V = \frac{4}{3}\,\pi\,a\,b\,c \]

Here \(a\), \(b\), and \(c\) are semi-axes (radii), not full diameters. If you have measured full axis lengths \(D_1\), \(D_2\), \(D_3\), then \(a = D_1/2\), \(b = D_2/2\), \(c = D_3/2\), and the formula can be written as:

\[ V = \frac{\pi}{6}\,D_1\,D_2\,D_3 \]

This second form is widely used in radiology and biology (for example in prostate or organ volume estimation) where three orthogonal diameters are measured from images.

Relationship with the volume of a sphere

A sphere of radius \(r\) is a special case of an ellipsoid with equal semi-axes: \(a = b = c = r\). Plugging this into the ellipsoid formula gives:

\[ V = \frac{4}{3}\,\pi\,r^3, \]

which is exactly the familiar expression for the volume of a sphere. Any deviation of the three semi-axes from equality represents an elongation or flattening of the sphere.

Solving for an unknown axis

The formula can be rearranged to solve for a missing semi-axis. Suppose \(V\), \(b\) and \(c\) are known and \(a\) is unknown:

\[ V = \frac{4}{3}\,\pi\,a\,b\,c \quad\Rightarrow\quad a = \frac{3V}{4\pi b c}. \]

Similarly, solving for \(b\) or \(c\) just cycles the roles of the axes. The calculator’s inverse mode performs these rearrangements and checks whether the inputs are physically consistent (for example, positive volume and positive axes).

Units and practical conversions

As long as all three semi-axes use the same length unit, the computed volume will be in the corresponding cubic unit:

mm → mm³ (often interpreted as microlitres μL)
cm → cm³ (numerically equal to mL)
m → m³ (1 m³ = 1,000 L)
in → in³
ft → ft³

The calculator automatically converts the result to cubic meters and liters when meaningful, so you can quickly compare geometric volumes with storage or flow capacities.

Example: ellipsoid volume from diameters

Suppose a roughly ellipsoidal tank has measured diameters \(D_1 = 4.0\ \text{m}\), \(D_2 = 3.0\ \text{m}\), and \(D_3 = 2.0\ \text{m}\).

Convert to semi-axes: \(a = 2.0\ \text{m}\), \(b = 1.5\ \text{m}\), \(c = 1.0\ \text{m}\).
Compute the product \(a b c = 2.0 \times 1.5 \times 1.0 = 3.0\ \text{m}^3\).
Apply the formula: \[ V = \frac{4}{3}\pi a b c = \frac{4}{3}\pi \times 3.0 \approx 12.57\ \text{m}^3. \]
Convert to liters: \(12.57\ \text{m}^3 \times 1000 \approx 12{,}570\ \text{L}.\)

Good practice when using ellipsoid volume approximations

Ensure that the shape is reasonably close to an ideal ellipsoid; strong irregularities can introduce bias.
Take measurements along orthogonal directions that truly represent the principal axes.
Be consistent with units: mixing centimeters and millimeters in the same product is a common error.
For safety-critical engineering or medical decisions, compare the ellipsoid estimate with alternative methods (numerical integration, voxel counting, or manufacturer data).

Related volume calculators

If you are working with other solids, these tools may also be helpful: