Ellipsoid – a useful shape
An ellipsoid is a surface that might be obtained by "squeezing" a typical ball. It's similar to the American football ball with smoothed corners. 🏈 What's interesting is all the cross-sections of an ellipsoid are in the shape of an ellipse.
We define ellipsoids with the use of semi-axes — line segments that start at the very center of the ellipsoid and finish at the point tangent with the surface (you can think of it the same way as you do about the radius of a circle). We can distinguish three types of semi-axes:
Based on an ellipsoid's cross section (ellipse):
- Semi-major axis (a) — the biggest one; and
- Semi-minor axis (b) — an axis at right angles to the semi-major axis.
3D modification:
- Third axis (c) — at right angles to the two preceding axes.
All three semi-axes meet at the center of the ellipsoid.
Why do we need the ellipsoid volume? 🤔 This shape is pretty common in nature. It's usually used in medicine, in order to estimate the volume of different organs, such as:
- Ovaries;
- Prostate; or
- Urinary bladder.
The ellipse itself is also used in calculating the movements of the planets.
How to use the ellipsoid calculator?
Our ellipsoid volume calculator is simple to use and consists of two main steps:
- Choose your unit system — metric (centimeters) or US/Imperial (inches).
- Find the lengths of all three semi-axes of your ellipsoid. All of them need to be at 90° (right angles) to each other.
- Enter the obtained values and enjoy your result! 🎉
We'll display the ellipsoid volume formula, as well as the solution — in multiple units your heart may desire!
How to calculate the volume of an ellipsoid
We can calculate the volume of an elliptical sphere with a simple and elegant ellipsoid equation:
where:
- a, b, and c are the lengths of all three semi-axes of the ellipsoid.
- π (pi) ≈ 3.14159265…
Example
Let's say we have an ellipsoid with semi-axes a = 5 cm, b = 3 cm, and c = 2 cm:
Ellipsoid formula
This section will show you how you can designate the ellipsoid using two different methods.
We need to use the Cartesian coordinate system in three dimensions (x, y, z). Then, we need to set the origin of the coordinate system (0, 0, 0) as the center of the ellipsoid.
Use the values of the semi-axes:
Find these three points in the coordinate system:
- (a, 0, 0)
- (0, b, 0)
- (0, 0, c)
These are the points of the surface that constitute the border of your ellipsoid.
Use the ellipsoid formula:
This equation is also useful if you need to find the value of any of the semi-axes.
Real-life applications
In wireless communication, we have a 3D elliptical region (an ellipsoid volume) between the transmitter antenna and the receiver antenna. This region is determined by the distance between the antennas and the frequency of the wireless wave. It is called the Fresnel zone.
The main ellipsoid volume, called the Fresnel zone, has to be kept free in order to secure proper wireless communication. Formulas for determining the space to be kept free are derived from the main ellipsoid equation.
Other real-life applications include:
- Medicine — estimating organ volumes (ovaries, prostate, bladder)
- Astronomy — modeling the shape of planets and moons
- Engineering — designing tanks, domes, and pressure vessels
- Physics — modeling atomic nuclei and molecular shapes
- Sports — American football, rugby ball design
Unit Systems
Metric System
Enter semi-axes in centimeters (cm). The calculator returns the volume in:
- cm³ — cubic centimeters
- m³ — cubic meters (1 m³ = 1,000,000 cm³)
- L — liters (1 L = 1,000 cm³)
US / Imperial System
Enter semi-axes in inches (in). The calculator returns the volume in:
- in³ — cubic inches
- ft³ — cubic feet (1 ft³ = 1,728 in³)
- US gal — US gallons (1 US gal = 231 in³)
FAQs
What is an ellipsoid?
An ellipsoid is a 3D shape where every cross-section is an ellipse. A sphere is a special case of an ellipsoid where all three semi-axes are equal (a = b = c).
What is the difference between a semi-axis and an axis?
An axis passes through the center from one side to the other. A semi-axis is half of that — from the center to the surface. If the full axis has length 10 cm, then the semi-axis is 5 cm.
What if all three semi-axes are equal?
If a = b = c = r, the ellipsoid becomes a sphere, and the formula reduces to the familiar sphere volume formula: V = (4/3) × π × r³.
What if two semi-axes are equal?
If two semi-axes are equal (e.g., a = b ≠ c), the shape is called a spheroid. A prolate spheroid (c > a) looks like a rugby ball; an oblate spheroid (c < a) looks like a flattened sphere.