Volume of an Ellipsoid Formula

Ellipsoid Volume Calculator

Ellipsoid – a useful shape

Based on an ellipsoid’s cross section (ellipse):

• Semi-major axis – the biggest one; and
• Semi-minor axis – an axis at right angles to the semi-major axis.

3D modification:

• Third axis is at right angles to the two proceeding axes.

All three semi-axes meet at the center of the ellipsoid

Why do we need the ellipsoid volume? 🤔 This shape is pretty common in nature. It’s usually used in medicine, in order to estimate the volume of different organs, such as:

• Ovaries;
• Prostate; or
• Urinary bladder.

How to use the ellipsoid calculator?

Real-life applications

Volume of an Ellipsoid

Volume of the Ellipsoid

The volume of the ellipsoid is the measurement of the ellipsoid that expresses the amount of three-dimensional space enclosed by a closed surface.

As we know that the equation of ellipsoid be (x²/a²) + (y²/b²) + (z²/c²) = 1 where a, b, c are the lengths of semi-axes of ellipsoid then volume can be calculated by the below formula-

Volume of Ellipsoid = (4/3) × π × a × b × c

The volume of Oblate Ellipsoid is

Volume of Oblate Ellipsoid = (4/3) × π × a × a × b

The volume of Prolate Ellipsoid is

Volume of Prolate Ellipsoid = (4/3) × π × a × b × b

Example:

Given the length of semi-axes are 5cm, 6cm, 4cm

So the volume of the ellipsoid is

V = (4/3) × π × a × b × c

= (4/3) × π × 5 × 6 × 4

= 430/3

= 160

Hence the volume of the ellipsoid is 160

Determining the volume of the ellipsoid

As we know that the equation of ellipsoid is

(x²/a²) + (y²/b²) + (z²/c²) = 1

Let us assume that -a ≤ x ≤ a

Now, we cut the ellipsoid with a plane parallel to the yz-plane

So, we get an ellipse

(y²/b²) + (z²/c²) = 1 – (x²/a²)

(y²/b²(1 – (x²/a²))) + (z²/c²(1 – (x²/a²) )) = 1

So the semiaxes are

p = b√(1 – (x²/a²)) and q = c√(1 – (x²/a²))

As we know that the area of ellipse is

A(x) =  πbc(1 – (x²/a²)) …..(1)

Now by using the formula of parent entry we calculate the volume of the ellipsoid

Sample Questions

Question 1: Find the volume of the ellipsoid if the lengths of semi-axes are 3cm, 4cm, 2cm.

Solution:

Given,

Lengths of semi axes of an ellipsoid a=3cm, b=4cm, c=2cm

Volume = (4/3) × π × a × b × c

= (4/3) × π × 3 × 4 × 2

= 32 × π

= 100.53 cm³

So, volume of ellipsoid with given measurements is 100.53cm³.

Question 2: Find the volume of the ellipsoid if the lengths of semi-axes are 5cm, 3cm, 2cm.

Solution:

Given,

Lengths of semi axes of an ellipsoid a = 5cm, b = 3cm, c = 2cm

Volume = (4/3) × π × a × b × c

= (4/3) × π × 5 × 3 × 2

= 40 × π

= 125.66 cm³

So, volume of ellipsoid with given measurements is 125.66cm³.

Question 3: Find the volume of the ellipsoid if the lengths of axes are 6cm, 4cm, 2cm.

Solution:

Given,

Lengths of axes of an ellipsoid are 6cm, 4cm and 2cm.

Length of semi axes = Length of axes/2

a = (6/2) = 3cm

b = (4/2) = 2cm

c = (2/2) = 1cm

Volume = (4/3) × π × a × b × c

= (4/3) × π × 3 × 2 × 1

= 8× π

= 25.13 cm³

So, volume of ellipsoid with given measurements is 25.13cm³.

Question 4: Find the volume of the ellipsoid if the lengths of axes are 12cm, 6cm, and 2cm.

Solution:

Given,

Lengths of axes of an ellipsoid are 12cm, 6cm and 2cm.

Length of semi axes = Length of axes/2

a = (12/2) = 6cm

b = (6/2) = 3cm

c = (2/2) = 1cm

Volume = (4/3) × π × a × b × c

= (4/3) × π × 6 × 3 × 1

= 24× π

= 75.4 cm³

So, the volume of ellipsoid with given measurements is 75.4cm³.

Question 5: Find the volume of ellipsoid if the equation is given as (x²/7²) + (y²/4²) + (z²/2²) = 1

Solution:

Given,

Equation of ellipsoid, (x²/7²) + (y²/4²) + (z²/2²) = 1

It is of form (x²/a²) + (y²/b²) + (z²/c²) = 1

From this we can derive lengths of semi axes.

a = 7

b = 4

c = 2

Volume = (4/3) × π × a × b × c

= (4/3) × π × 7 × 4 × 2

= (224/3) × π

= 234.57 cm³

So, the volume of ellipsoid with given measurements is 234.57cm³.

Volume of an Ellipsoid Formula

An ellipsoid is a closed quadric surface that is a three-dimensional analogue of an ellipse. The standard equation of an ellipsoid centred at the origin of a Cartesian coordinate system. The spectral theorem can again be used to obtain a standard equation akin to the definition given above.

Where,
a = r₁ = Radius of the ellipsoid of axis 1
b = r₂ = Radius of the ellipsoid of axis 2
c = r₃ = Radius of the ellipsoid of axis 3

Volume of an Ellipsoid Formula Solved Example

Example: The ellipsoid whose radii are given as a = 9 cm, b = 6 cm and c = 3 cm. Find the volume of an ellipsoid.

Solution:

Given,
Radius (a) = 9 cm
Radius (b) = 6 cm
Radius (c) = 3 cm

Using the formula:

How to Use the Ellipsoid Volume Calculator?

The procedure to use the ellipsoid volume calculator is as follows:

Step 1: Enter the radius values in the input field

Step 2: Now click the button “Solve” to get the volume

Step 3: Finally, the volume of ellipsoid will be displayed in the output field

What is Meant by Ellipsoid Volume?

In Mathematics, an ellipsoid is a surface which is obtained from the shape “sphere” by deforming it in terms of directional scalings. In other words, an ellipsoid is a squashed sphere. The three pairwise perpendicular axes of symmetry intersect at the centre of symmetry is called the centre of the ellipsoid. If the two axes have the same symmetry, then the ellipsoid is called the spheroid. If a, b and c are the radius, then the volume of the ellipsoid formula is given by

The volume of the ellipsoid, V = (4/3)πabc cubic units.

Example:

Find the volume of the ellipsoid whose radii are 21 cm, 15 cm and 2 cm.

Solution:

Given:

a= 21 cm

b =15 cm

c =2 cm

We know that the volume of the ellipsoid is (4/3)πabc cubic units

Now, substitute the values in the formula, we get

V = (4/3)π(21)(15)(2) cubic units

V = 2640 cm³.

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