Volume of an object is the amount of space occupied by the three dimensional object i.e., the quantity of three dimensional space present in a solid object. It is measured in terms of cubic units.
Volume is used to measure how much space is occupied or covered by a shape. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas. Volume can be found for geometrical shapes like cone, sphere, cylinder, prism, ellipsoid, tetrahedron etc., Volumes of a cone, sphere and cylinder of the same radius and height are in the ratio 1 : 2 : 3.

## Volume Formulas

Below are listed volume formulas for shapes of geometry.

 Geometrical Shape Volume Formula (in cubic units) Description of the variables Cube l$^{3}$ l = length Cuboid l $\times$ b $\times$ h l = length, b = breadth, h = height Sphere $\frac{4}{3}$ $\pi$ r$^{3}$ r = radius Cylinder $\pi$ $r^{2}$h r = radius, h = height Cone $\frac{1}{3}$ $\pi$ $r^{2}$h r = base radius, h = height Hemisphere $\frac{2}{3}$ $\pi$ $r^{3}$ r = radius Prism B.h B = area of the base, h = height Pyramid $\frac{1}{3}$ B.h B = area of the base, h = height of pyramid Tetrahedron $\frac{\sqrt{2}}{12}$ a$^{3}$ a = edge length Ellipsoid $\frac{4}{3}$ $\pi$ abc a, b, c = semi axes of ellipsoid Hexagonal pyramid a $\times$ b $\times$ h a = apothem length of the pyramid,   b = base length of the pyramid h = height of the pyramid Pentagonal pyramid $\frac{5}{6}$ a$\times$ b$\times$ h a = apothem length of the pyramid,  b = base length of the pyramid h = height of the pyramid Triangular pyramid $\frac{1}{6}$ a $\times$ b$\times$ h a = apothem length of the pyramid,    b = base length of the pyramid h = height of the pyramid Square pyramid $\frac{1}{3}$ b$^{2}$ h b = base length of the pyramidh = height of the pyramid

## Calculate Volume

### Solved Examples

Question 1: Find the volume of the cylinder with a radius of 5 cm and height 8 cm
Solution:

Height = 8 cm
Volume of a cylinder is $\pi$ r$^{2}$ h
V = 3.14 $\times$ 25 $\times$ 8
= 628

Therefore, volume of the cylinder for the given data is 628 cm$^{3}$.

Question 2: David has a rectangular garden pond 5 m long and 3 m wide. How many litres of water does he need to fill it to a depth of 45 cm.
Solution:

Dimensions of rectangular garden pond = 5 m x 3 m x 45 cm
Convert all the given units in centimeters.

Dimensions = 5 m x 3 m x 45 cm = 500 cm x 300 cm x 45 cm
Now volume of cuboid = 500 $\times$ 300 $\times$ 45
= 6750000 cm$^{3}$
As 1 litre = 1000 cm$^{3}$,

$\frac{6750000}{1000}$ = 6750

Therefore, David will need 6750 litres of water to fill rectangular pond to a depth of 45 cm.

Question 3: Find the volume of a sphere of radius 24 m.
Solution:

Formula for volume of sphere is given by

$\frac{4}{3}$ $\pi$ r$^{3}$

= $\frac{4}{3}$ $\times$ 3.14 $\times$ (24)$^{3}$

= 57876.48
Therefore, the volume of the sphere is 57876.48 m$^{3}$

Question 4: Find the volume of a pyramid with square base of side 8 cm and a height of 12 cm.
Solution:

Height = 12 cm
Area of base = Side$^{2}$ (l$^2$) = 8$^{2}$ = 64

The formula for volume of pyramid is $\frac{1}{3}$ * l$^{2}$ $\times$ h

= $\frac{1}{3}$ $\times$ 64 $\times$ 12

= 256
Therefore, the volume of the pyramid for the given data is 256 cm$^{3}$.

Question 5: Find the volume of a cone of radius 8 cm and height 13 cm?
Solution:

Height = 13 cm
The formula for volume of cone is $\frac{1}{3}$ $\times$ $\pi$ r$^{2}$ $\times$ h
= 870.83
Therefore, the volume of a cone is 870.83 cm$^{3}$

Question 6: Find the volume of a triangular pyramid of apothem length 9 cm, base length 12 cm and height 15 cm ?
Solution:

Given,
a = 9 cm
b = 12 cm
h = 15 cm

Volume of a triangular pyramid
= $\frac{1}{6}$ abh

= $\frac{1}{6}$ * 9 * 12 * 15

= 270
Therefore, the volume of a triangular pyramid is 270cm$^{3}$.

Question 7: Find the volume of a pentagonal pyramid of apothem length 7 cm, base length 8 cm and height 10 cm ?
Solution:

Given,
a = 7 cm
b = 8 cm
h = 10 cm

Volume of a pentagonal pyramid

= $\frac{5}{6}$ abh

= 466.67

Therefore, the volume of a pentagonal pyramid for the given data is 466.67 cm$^{3}$