Ellipsoid volume calculator

What is an ellipsoid?

An ellipsoid is a three-dimensional geometric surface that is the three-dimensional analog of an ellipse. Simply put, an ellipsoid exhibits symmetry in all directions and looks like an elongated or flattened sphere. Mathematically, it is defined as the set of points $(x, y, z)$ such that:

where $a$, $b$, and $c$ are the lengths of the semi-principal axes of the ellipsoid. If all three axes are equal, the ellipsoid becomes a perfect sphere. For more information on spheres, see our sphere volume calculator.

Formula to calculate the volume of an ellipsoid

The formula used to calculate the volume $V$ of an ellipsoid is given by:

Where:

• $V$ represents the volume of the ellipsoid,
• $a$, $b$, and $c$ are the semi-principal axes of the ellipsoid,
• $\pi$ is a constant approximately equal to 3.14159.

This formula shows that the volume of an ellipsoid is directly proportional to the product of its semi-principal axes and the constant $\pi$.

Examples of ellipsoid volume calculations

Example 1

Calculate the volume of an ellipsoid with semi-principal axes lengths $a = 3$, $b = 4$, and $c = 5$.

Using the formula:

Substitute in the given values:

Thus, the volume is approximately $251.33$ cubic units.

Example 2

Calculate the volume of a spheroid, a special kind of ellipsoid, with axes $a = 5$, $b = 5$, and $c = 2$.

Using the formula:

Substitute in the given values:

So, the volume is approximately $209.44$ cubic units.

Example 3

Find one of the semi-principal axes of an ellipsoid, if the volume and the other two semi-principal axes are known.

Let $V = 1000$ cubic units, $a = 5$ and $b = 6$.

Using the formula:

Thus, $c \approx 7.96$.

Practical applications of ellipsoid volume

Understanding the volume of ellipsoids is not just a mathematical exercise but also has numerous practical applications in various fields:

• Physics and Astronomy: The shape and volume of planets, stars, and other celestial bodies are often modelled as ellipsoids.
• Biology: Many biological cells and microorganisms are approximately ellipsoidal, and their volume calculations are essential in biological studies.
• Engineering: Design and analysis of structures and components like pressure vessels or storage tanks often involve ellipsoidal shapes.

Historical insights on ellipsoids

The study of ellipsoids can be traced back to ancient Greek mathematicians who explored the properties of ellipses and extended these properties into three dimensions. The formulas we use today are built upon centuries of mathematical development.

Friedrich Wilhelm Bessel, in the 19th century, made significant contributions to understanding ellipsoids while trying to measure the shape of Earth, which is slightly ellipsoidal rather than a perfect sphere.

Frequently asked questions

Why use an ellipsoid volume calculator?

The calculator simplifies the process of finding the volume of an ellipsoid by automating the calculation process. It ensures accuracy and saves time, especially in professional or academic settings where multiple calculations might be necessary.

How to calculate the volume of an ellipsoid?

To calculate the volume of an ellipsoid, multiply $\frac{4}{3}\pi$ by the lengths of the three semi-principal axes ($a$, $b$, $c$).

Are ellipsoids always symmetrical?

Ellipsoids are characterized by their symmetry relative to their three orthogonal axes. However, they do not need to have equal symmetry across all axes, resulting in diverse shapes such as prolate and oblate spheroids.

Can volume calculators be used for celestial bodies modelled as ellipsoids?

Yes, many celestial bodies like planets and asteroids can be considered ellipsoids, and their volume can be calculated to better understand their mass and gravitational force.