Volume of an Ellipsoid

Volume of an Ellipsoid Formula: An ellipsoid can be called a 3D equivalent of an ellipse. It can be derived from a sphere by contorting it using directional scaling or, more broadly, an interpolation conversion. An ellipsoid is evenly spaced at three coordinate axes that intersect at the center. 

In this article, we will discuss the volume of an ellipsoid, and provide solved examples on it.

The above picture shows an ellipsoid, its three semi-axes denoted a, b, and c are also shown. Ellipsoids occur in nature in the shape of watermelons, as well as the female reproductive organs and male urinary bladder. The study of the ellipsoid volume is necessary as it helps doctors calculate the volume of ovaries and urinary bladders.

What is an Ellipsoid?

An ellipsoid is a three-dimensional geometric shape resembling a flattened sphere or an elongated sphere. It is defined as the set of all points in three-dimensional space whose distances from a fixed point (called the center) have a certain relationship.

Volume of Ellipsoid Formula

[Tex]V = \frac{4}{3}\pi abc [/Tex]

where a, b and c are the semi- axes of the given ellipsoid.

Solved Examples on Volume of an Ellipsoid

Problem 1. Find the volume of an ellipsoid with 5, 6, and 11 cm radii.

Solution:

Given: a = 5 cm, b = 6 cm, c = 11 cm

Since, [Tex]V = \frac{4}{3}\pi abc [/Tex]

= 4 × 3.14 × 5 × 6 × 11/3

V = 1382.6 cm³

Problem 2. Find the volume of an ellipsoid whose radii are 3, 4 and 7 cm.

Solution:

Given: a = 3 cm, b = 4 cm, c = 7 cm

Since, V = [Tex]\frac{4}{3}\pi abc [/Tex]

= 4 × 3.14 × 3 × 4 × 7/3

V = 351.68 cm³

Problem 3. Find the volume of an ellipsoid whose radii are 4, 7 and 11 cm.

Solution:

Given: a = 4 cm, b = 7 cm, c = 11 cm

Since, V = [Tex]\frac{4}{3}\pi abc [/Tex]

= 4 × 3.14 × 4 × 7 × 11/3

V = 1289.49 cm³

Problem 4. Find the volume of an ellipsoid whose radii are 8, 6 and 14 cm.

Solution:

Given: a = 8 cm, b = 6 cm, c = 14 cm

Since, V = [Tex]\frac{4}{3}\pi abc [/Tex]

= 4 × 3.14 × 8 × 6 × 14/3

V = 2813.44 cm³

Problem 5. Find the volume of an ellipsoid whose radii are 2, 6 and 8 cm.

Solution:

Given: a = 2 cm, b = 6 cm, c = 8 cm

Since, V = [Tex]\frac{4}{3}\pi abc [/Tex]

= 4 × 3.14 × 2 × 6 × 8/3

V = 401.92 cm³

Volume of an Ellipsoid Practice Problems

1. Given an ellipsoid with semi-axis lengths a = 3 cm, b = 4 cm, and c = 5 cm, calculate its volume.

2. If the volume of an ellipsoid is 4𝜋/3 cubic meters and its semi-axis lengths are a = 2 meters, b = 3 meters, and c = 4 meters, find the surface area of the ellipsoid.

3. An ellipsoid has a volume of 32𝜋/3 cubic units. If two of its semi-axis lengths are 2 units each, find the length of the third semi-axis.

4. Find the volume of an ellipsoid whose semi-axis lengths are all equal to 6 inches.

5. An ellipsoid has a volume of 1000 cubic centimeters. If its semi-axis lengths are in the ratio 3:4:5, find the lengths of its semi-axes.