# Volume of an Ellipsoid Formula

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## Ellipsoid Volume Calculator

Are you looking for an ellipsoid volume calculator? You’ve just found the perfect place! We’re going to compute the volume of an ellipsoid and give you a step-by-step solution so that you can learn how to do it yourself.

Follow the article below and discover the ellipsoid volume formula, the ellipsoid shape properties, and other useful pieces of information.

## Ellipsoid – a useful shape

An ellipsoid is a surface that might be obtained by “squeezing” a typical ball. It’s similar to the American football ball with smoothed corners. 🏈 What’s interesting is all the cross-sections of an ellipsoid are in a shape of an ellipseWe define ellipsoids with the use of semi-axes – line segments that start at the very center of the ellipsoid and finish at the point tangent with the surface (you can think of it the same way as you do about the radius of a circle). We can distinguish three types of semi-axes:.

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

## How to use the ellipsoid calculator?

Our ellipsoid volume calculator is simple to use, and consists of two main steps:

1. Find the lengths of all three axes of your ellipsoid.

All of them need to be at 90° (right angles) to each other.

2. Enter the obtained values and enjoy your result! 🎉

We’ll display the ellipsoid volume formula, as well as our solution – in all the possible units your heart may desire!

## How to calculate the volume of an ellipsoid?

We can calculate the volume of an elliptical sphere with a simple and elegant ellipsoid equation:

`Volume = 4/3 × π × A × B × C`,

where:

• A, B, and C are the lengths of all three semi-axes of the ellipsoid.

## Ellipsoid formula

This section will show you how you can designate the ellipsoid using two different methods.

We need to use the Cartesian coordinate system in three dimensions (x, y, z). Then we need to set the origin of the coordinate system (0, 0, 0) as the center of the ellipsoid.

1. Use the values of the semi-axes

Find these three points in the coordinate system:

• (A, 0, 0)
• (0, B, 0)
• (0, 0, C)

These are the points of the surface that constitute the border of your ellipsoid.

2. Use the ellipsoid formula

`1 = (x²/A²) + (y²/B²) + (z²/C²)`

This equation is also useful if you need to find the value of any of the semiaxes.

## Real-life applications

In wireless communication, we have a 3D elliptical region (an ellipsoid volume) between the transmitter antenna and the receiver antenna. This region is determined by the distance between the antennas and the frequency of the wireless wave. It is called Fresnel zone and looks like this:

## 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,

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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