[Maths Class Notes] on Ellipsoid Pdf for Exam

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 . It is a free online tool that displays the surface area and volume of ellipsoid for the given radii.

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.