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.
Equation of Ellipsoids
When two axes, such as m and n, are in equivalency and distinct from the third, o, the ellipsoid is referred to as a spheroid or an ellipsoid of revolution. An ellipse is rotated about one of its axes to generate an ellipsoid shape.
The spheroid will be oblate if m and n are more than o; if they are smaller, the surface will be prolate. Having stated that, a classic equation of such an ellipsoid is x²/m² + y²/n² + z²/o² = 1, assuming that m, n, and o are the primary semiaxes. When m = n = o, there is a special case where the surface is a sphere and the bisection through which any plane passes is a circle.
V = \frac{4}{3}\pi abc
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, V = \frac{4}{3}\pi abc
= 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 = \frac{4}{3}\pi abc
= 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 = \frac{4}{3}\pi abc
= 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 = \frac{4}{3}\pi abc
= 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 = \frac{4}{3}\pi abc
= 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 4?/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 32?/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.
Conclusion of Ellipsoid
An ellipsoid is a three-dimensional geometric shape that generalizes the concept of an ellipse to higher dimensions. It is defined as the set of all points for which the sum of the distances to the foci is constant. Understanding the geometry and properties of ellipsoids is essential in various fields, including physics, engineering, and astronomy, where they model phenomena such as planetary shapes and stress distributions.