Standard Equation of an Ellipsoid with Formula and Solved Examples
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 Three Dimensional Geometry
1. What is an ellipsoid in mathematics?
An ellipsoid is a three-dimensional surface obtained by stretching or compressing a sphere along its coordinate axes. It is defined as the set of all points (x, y, z) that satisfy the equation:
x²/a² + y²/b² + z²/c² = 1
where a, b, and c are the semi-axes along the x, y, and z directions. If all three are equal, the ellipsoid becomes a sphere; if they are different, it forms an elongated or flattened shape.
2. What is the standard equation of an ellipsoid?
The standard equation of an ellipsoid centered at the origin is x²/a² + y²/b² + z²/c² = 1.
- a, b, and c are the semi-axis lengths.
- If a = b = c, the surface is a sphere.
- If two are equal (a = b ≠ c), it is a spheroid.
3. What is the formula for the volume of an ellipsoid?
The volume of an ellipsoid is given by V = (4/3)πabc.
- a, b, and c are the semi-axes.
- This formula generalizes the sphere volume formula.
V = (4/3)π(2)(3)(4) = 32π cubic units.
4. How do you find the intercepts of an ellipsoid?
The intercepts of an ellipsoid are found by setting two variables to zero and solving for the third. For x²/a² + y²/b² + z²/c² = 1:
- x-intercepts: (±a, 0, 0)
- y-intercepts: (0, ±b, 0)
- z-intercepts: (0, 0, ±c)
5. What is the difference between a sphere and an ellipsoid?
The main difference is that a sphere has all equal radii, while an ellipsoid can have three different semi-axes.
- Sphere: x² + y² + z² = r²
- Ellipsoid: x²/a² + y²/b² + z²/c² = 1
6. What are the types of ellipsoids?
Ellipsoids are classified based on their semi-axes as triaxial ellipsoids and spheroids.
- Triaxial ellipsoid: a ≠ b ≠ c
- Oblate spheroid: a = b > c (flattened at poles)
- Prolate spheroid: a = b < c (elongated shape)
7. How do you graph an ellipsoid?
To graph an ellipsoid, plot its intercepts and sketch the smooth curved surface connecting them. Steps:
- Write the equation in standard form.
- Identify semi-axes a, b, and c.
- Mark intercepts (±a, 0, 0), (0, ±b, 0), (0, 0, ±c).
- Draw elliptical cross-sections in coordinate planes.
8. Can you give an example of solving an ellipsoid equation?
Yes, for example, determine whether (1, 1, 1) lies on the ellipsoid x²/4 + y²/9 + z²/16 = 1. Substitute the values:
- (1²/4) + (1²/9) + (1²/16)
- = 1/4 + 1/9 + 1/16
36/144 + 16/144 + 9/144 = 61/144 ≠ 1.
Since the sum is not 1, the point does not lie on the ellipsoid.
9. What are the properties of an ellipsoid?
An ellipsoid has several key geometric properties related to symmetry and cross-sections.
- It is symmetric about the coordinate axes.
- All cross-sections parallel to coordinate planes are ellipses.
- It is a closed and bounded 3D surface.
- Its center is at the origin in standard form.
10. Where are ellipsoids used in real life?
Ellipsoids are used in astronomy, physics, and engineering to model real-world shapes.
- The Earth is approximately an oblate spheroid.
- Planetary orbits and gravitational models use ellipsoidal geometry.
- Engineering designs use ellipsoidal tanks and reflectors.
