Volume of Ellipsoid Calculator

Q: What is the formula for the volume of an ellipsoid?

The volume of an ellipsoid is calculated using V = (4/3)πabc, where a, b, and c are the three semi-axes. This is a generalization of the sphere volume formula, where a sphere is a special case with a = b = c = r.

Calculate the volume of an ellipsoid with step-by-step solution, interactive 3D diagram, surface area estimation, and unit conversions. Supports all common measurement units.

Free to useNo sign-up requiredUpdated Feb 2026

Volume of Ellipsoid CalculatorTry it now — free ▼

★ Quick Examples:

📐 General (5,3,4) 🌍 Earth ⚽ Sphere (r=7) 🏈 Prolate (10,5,3) 🥚 Egg shape

Semi-axes: a, b, c from center

a Semi-axis a

b Semi-axis b

c Semi-axis c

Unit

Precision

Embed Volume of Ellipsoid Calculator Widget

About Volume of Ellipsoid Calculator

The Volume of Ellipsoid Calculator computes the exact volume of any ellipsoid using the formula $ V = \frac{4}{3}\pi abc $, with step-by-step solutions, interactive 3D visualization, surface area approximation, cross-sectional areas, and automatic shape classification. Whether you are studying geometry, working in engineering, or modeling natural shapes, this tool provides comprehensive ellipsoid analysis.

Ellipsoid Volume Formula

The volume of an ellipsoid with semi-axes a, b, and c is given by:

Volume of an Ellipsoid

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

Where:

V = volume of the ellipsoid
a = semi-axis along the x-direction
b = semi-axis along the y-direction
c = semi-axis along the z-direction
π ≈ 3.14159265...

This formula is a generalization of the sphere volume formula $ V = \frac{4}{3}\pi r^3 $, where a sphere is the special case with a = b = c = r.

Types of Ellipsoids

Ellipsoids are classified based on the relationship between their semi-axes:

🟡

Sphere

All axes equal: a = b = c. Perfect symmetry in all directions.

🏈

Prolate Spheroid

Two short axes equal, one long. Shaped like a rugby ball or egg.

🌍

Oblate Spheroid

Two long axes equal, one short. Shaped like Earth or a lentil.

💎

Scalene Ellipsoid

All three axes different. The most general ellipsoid form.

Surface Area of an Ellipsoid

Unlike volume, there is no simple closed-form expression for the surface area of a general ellipsoid. It requires elliptic integrals. This calculator uses Knud Thomsen's approximation, which is accurate to about 1.06%:

Surface Area Approximation (Knud Thomsen)

$$S \approx 4\pi \left(\frac{a^p b^p + a^p c^p + b^p c^p}{3}\right)^{1/p}$$

Where p ≈ 1.6075. For a sphere (a = b = c = r), this reduces to the exact formula $ S = 4\pi r^2 $.

How to Calculate Ellipsoid Volume

Measure the semi-axes: Determine the three semi-axes a, b, and c. A semi-axis is half the full diameter along each direction.
Enter values: Input the three semi-axis lengths, select your measurement unit, and choose decimal precision.
Review results: Get the volume, surface area, shape classification, cross-sectional areas, and a detailed step-by-step breakdown with 3D visualization.

Real-World Applications

🌍 Geodesy & Geography

Earth is modeled as an oblate spheroid (WGS84 ellipsoid) with equatorial radius 6378.137 km and polar radius 6356.752 km, used in GPS and mapping.

🔭 Astronomy

Many planets, moons, and asteroids are ellipsoidal. Calculating their volumes helps determine mass and density.

🏥 Medical Imaging

Organs and tumors are often approximated as ellipsoids in CT/MRI scans to estimate volume for diagnosis and treatment planning.

🏗️ Engineering

Ellipsoidal tanks, domes, and structural components require volume calculations for capacity planning and material estimation.

🧬 Biology

Cells, eggs, and microorganisms often have ellipsoidal shapes. Volume calculation helps in biological modeling and research.

🎮 Computer Graphics

Ellipsoids are fundamental bounding volumes and collision detection primitives in 3D rendering and physics engines.

Comparison with Other 3D Shapes

Shape	Volume Formula	Relationship to Ellipsoid
Sphere	$ \frac{4}{3}\pi r^3 $	Special case: a = b = c = r
Prolate Spheroid	$ \frac{4}{3}\pi a^2 c $	Special case: a = b < c
Oblate Spheroid	$ \frac{4}{3}\pi a^2 c $	Special case: a = b > c
Cylinder	$ \pi r^2 h $	Different shape; circumscribes ellipsoid when r = a, h = 2c
Rectangular Box	$ 2a \times 2b \times 2c $	Bounding box has volume $ 8abc $; ratio = $ \frac{\pi}{6} \approx 0.5236 $

Semi-Axis vs Full Axis

The volume formula uses semi-axes (half-lengths), not full diameters. If you have the full diameter measurements:

Semi-axis a = (full diameter along x) / 2
Semi-axis b = (full diameter along y) / 2
Semi-axis c = (full diameter along z) / 2

Frequently Asked Questions

What is the formula for the volume of an ellipsoid?

The volume of an ellipsoid is calculated using $ V = \frac{4}{3}\pi abc $, where a, b, and c are the three semi-axes. This is a generalization of the sphere volume formula, where a sphere is a special case with a = b = c = r.

What is the difference between an ellipsoid and a spheroid?

A spheroid is a special case of an ellipsoid where two of the three semi-axes are equal. A prolate spheroid (like a rugby ball) has two shorter equal axes, while an oblate spheroid (like Earth) has two longer equal axes. A general (scalene) ellipsoid has three different semi-axes.

What is the surface area of an ellipsoid?

Unlike volume, there is no simple closed-form formula for the surface area of a general ellipsoid. This calculator uses Knud Thomsen's approximation with accuracy within about 1.06%.

How is an ellipsoid used in real life?

Ellipsoids are used in geodesy (Earth is an oblate spheroid), astronomy (many celestial bodies are ellipsoidal), medical imaging (modeling organs), architecture, physics, computer graphics, and engineering. The WGS84 reference ellipsoid is the standard model for GPS coordinates.

What is the difference between semi-axis and full axis?

A semi-axis is half the length of a full axis. An ellipsoid with semi-axes a, b, c has full diameters of 2a, 2b, and 2c along each direction. The volume formula uses semi-axes (half-lengths), not full diameters.

Additional Resources

Reference this content, page, or tool as:

"Volume of Ellipsoid Calculator" at https://MiniWebtool.com/volume-of-ellipsoid-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Feb 13, 2026

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.