How to Find the Volume and Equation of an Ellipsoid
In geometry when we are to define an Ellipsoid, we say that it is a closed surface whose all plane cross-sections are either ellipses or circles. An ellipsoid is symmetrical at around three mutually perpendicular axes which bisect at the centre. The surface area of the ellipsoid, as well as the Ellipsoid Volume, can also be calculated using the online calculator available at Vedantu. It is a free online tool that displays the surface area and volume of ellipsoid for the given radii.
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Ellipsoid Equation
If two axes are in equivalence, say m = n, and different from the 3rd i.e., o, then the ellipsoid is said to be an ellipsoid of revolution or spheroid. The ellipsoid shape formed is by revolving an ellipse around one of its axes. If m and n are greater than o, the spheroid will be oblate; if lesser, the surface will be a prolate spheroid. Having said that, If m, n and o are the principal semiaxes, a standard equation of such an ellipsoid is x²/m² + y²/n² + z²/o² = 1. A unique case occurs when m = n = o: then the surface is a sphere, and the bisection with any plane crossing through it is a circle.
Volume of an Ellipsoid Formula
An ellipsoid is a closed quadric surface which is a 3-D analogue of an ellipse. The standard equation of momental ellipsoid centered at the origin of a Cartesian coordinate plane. The spectral theorem can again be used in order to acquire a standard equation akin to the explanation given above.
The Ellipsoid volume formula is given below:
V = 4/3 π m n o
or the formula can also be written as:
V = 4/3 π r1 r2 r3
Where,
M = r1 = Radius of the axis 1 of the ellipsoid
N = r2 = Radius of the axis 2 of the ellipsoid
O = r3 = Radius of the axis 3 of the ellipsoid
Solved Examples on Volume of an Ellipsoid
Example:
The Ellipsoid Which Has a Radii are Given as M = 12 cm, N = 9 cm and o = 4 cm. Find the Volume of an Ellipsoid.
Solution:
Given,
Radius (a) = 12 cm
Radius (b) = 9 cm
Radius (c) = 4 cm
Using the formula: V = 4/3 π a b c
V = 4/3 × 3.14 × 12 × 9 × 4
V = 1808.64 cm3
Example:
Evaluate the Volume of the Ellipsoid Whose Radii are 8 cm, 5 cm and 2 cm.
Solution:
Given:
m = 8 cm
n = 5 cm
o = 2 cm
We are aware that the volume of the ellipsoid is (4/3) π m n o cubic units
Now, plug the values into the formula, we obtain
V = (4/3) π (8)(5)(2) cubic units
V = (4/3) 3.14* (8)(5)(2)
V = 334.94 cm3.
Fun Facts
Ellipse, a closed curve, the bisection of a right circular cone and a plane which is not parallel to the base, the axis, or an element of the cone.
Another definition of an ellipsoid is that it is the locus of points for which the sum of their distances from two the foci (fixed points) is constant.
Ellipsoid can be defined as the path of a point moving in a plane in such a way that the ratio of its distances from the focus (certain point) and the directrix(fixed straight line) is a constant lesser than one.
Any such path consists of this same property in terms of the 2nd fixed point and a 2nd fixed line, and ellipses often are considered as having two foci and two directrixes.
The proportion of distances, termed as the eccentricity, is the discriminant (q.v.; of a general equation that denotes all the conic sections.
The shorter the distance between the foci, the lesser is the eccentricity and the more closely the ellipse represents a circle.
Isaac Newton anticipated that due to the Earth’s rotation, the shape of its axis must be an ellipsoid instead of spherical, and cautious measurements confirmed his anticipation.
With increased appropriacy in measurements became possible, further deviations from the elliptical shape were discovered.
FAQs on Ellipsoid in Maths: Formula, Properties & Real-World Uses
1. What is an ellipsoid in mathematics?
An ellipsoid is a three-dimensional surface that is a 3D analogue of an ellipse. It can be visualized as a sphere that has been stretched or compressed along its three perpendicular axes. The standard equation for an ellipsoid centred at the origin is (x²/a²) + (y²/b²) + (z²/c²) = 1, where 'a', 'b', and 'c' are the lengths of the semi-axes along the x, y, and z axes, respectively.
2. What is the formula for the volume of an ellipsoid?
The volume of an ellipsoid is calculated using a formula that involves its three semi-axes. If the lengths of the semi-axes are 'a', 'b', and 'c', the volume (V) is given by the formula: V = (4/3)πabc. Notice that if a = b = c, the ellipsoid becomes a sphere, and the formula simplifies to the familiar volume of a sphere, V = (4/3)πr³.
3. What are the different types of ellipsoids?
Ellipsoids are generally classified based on the relative lengths of their three semi-axes (a, b, c):
- Scalene Ellipsoid: This is the most general type, where all three semi-axes have different lengths (a ≠ b ≠ c).
- Spheroid (or Ellipsoid of Revolution): This type is formed by rotating an ellipse about one of its axes. There are two kinds of spheroids:
- Oblate Spheroid: Formed by rotating an ellipse about its minor axis. It is flattened at the poles, like the shape of the Earth (a = b > c).
- Prolate Spheroid: Formed by rotating an ellipse about its major axis. It is elongated, like a rugby ball or an American football (a = b < c).
4. What are some real-world examples of an ellipsoid shape?
The ellipsoid shape appears in many real-world objects, both natural and man-made. Some common examples include the shape of planets like Earth (which is an oblate spheroid), the general shape of an egg (though it's technically an ovoid), an American football or a rugby ball (which are prolate spheroids), and even the shape of some pills or candies.
5. How is an ellipsoid different from a sphere or an ovoid?
While they may look similar, these shapes have distinct mathematical properties. A sphere is a special case of an ellipsoid where all three semi-axes are equal (a=b=c). An ellipsoid is more general, allowing for different lengths of its three axes. An ovoid, or egg-shape, is different because it is not perfectly symmetrical across its wider middle; one end is more pointed than the other. An ellipsoid, in contrast, has three mutually perpendicular axes of symmetry.
6. Why is the Earth often described as an oblate ellipsoid and not a perfect sphere?
The Earth is described as an oblate ellipsoid because of its rotation. The centrifugal force generated by the Earth spinning on its axis causes it to bulge at the equator and flatten at the poles. As a result, the Earth's equatorial diameter is slightly larger than its polar diameter. A perfect sphere would have the same diameter in all directions, so the oblate ellipsoid is a more accurate mathematical model of our planet's shape.
7. What is the standard equation of an ellipsoid centred at the origin?
The standard equation of an ellipsoid with its centre at the origin (0, 0, 0) and its axes aligned with the coordinate axes is given by: (x²/a²) + (y²/b²) + (z²/c²) = 1. In this equation, 'a', 'b', and 'c' represent the lengths of the semi-axes along the x-axis, y-axis, and z-axis, respectively. These values determine the ellipsoid's size and shape.
8. How does the equation of an ellipsoid change if its centre is not at the origin?
If the centre of the ellipsoid is shifted from the origin to a new point (h, k, l), the standard equation is modified to reflect this translation. The new equation becomes: ((x-h)²/a²) + ((y-k)²/b²) + ((z-l)²/c²) = 1. This form is crucial for describing the position and orientation of ellipsoids in three-dimensional space that are not located at the origin.
9. Is there a simple formula for the surface area of an ellipsoid?
Unlike its volume, there is no single, simple formula to calculate the exact surface area of a general (scalene) ellipsoid using only elementary functions. The calculation is quite complex and involves special mathematical functions known as elliptic integrals. However, for spheroids (oblate or prolate), exact formulas exist, and for scalene ellipsoids, various approximation formulas are used in practice.
