# Volume of a Cylinder

The volume of a cylinder is the space occupied by the cylinder in the three-dimensional plane. It is the measure of the capacity of a cylinder or the number of unit cubes that can be fit into it. It is measured in cubic units such as m3, cm3, mm3, ft3.

Calculation of the volume of a cylinder is useful for designing cylindrical objects such as:

• Cylindrical water tanks and wells
• Perfume and chemical bottles
• Cylindrical containers and pipes

This article will deal with how to find the volume of a cylinder.

## Formulas

It is the product of the area of the circular base and the height of the cylinder. The general formula to get the volume of a cylinder is:

Volume (V) = A × h, here A = area of the circular base, h = height

Let us now learn the formulas for each of the 4 types of cylinders.

Derivation

As a cylinder is nothing but a set of circular discs stacked one upon the other. Now, if we find the space occupied by each of these discs and add them up we will get the volume of the cylinder.

Let us assume that the circular discs stack up to the height of ‘h’. Now, the volume of cylinder will be the product of the base area of the disc and the height.

Thus,

Volume (V) = Area of the circular base × height

Since,

Area of the circular bases = πr2

Height = h

Volume (V) = A × h

V = πr2 × h

V = πr2h

Thus, the equation to find the volume of a cylinder is V = πr2h

### Right Circular Cylinder

The formula to calculate the volume of a cylinder is given below. It is also the basic formula used to calculate the volume of a cylinder in general.

Thus, the volume is directly proportional to the height and square of its radius.

Let us solve some examples to understand the concept better.

Find the exact volume of a right circular cylinder of radius 40 cm and height 80 cm. Use π = 3.141.

Solution:

As we know,
Volume (V) = πr2h, here π = 3.141, r = 40 cm, h = 80 cm
= 3.141 × 40 × 40 × 80
= 4020480 cm3

The volume of a cylinder is 660 m3, and the radius of the base is 4 m. Calculate the height of the tank.

Solution:

As we know,
Volume (V) = πr2h, here V = 660 m3, π = 3.141, r = 4 m
660 = 3.141 × 4 × 4 × h
h = 660/3.141 × 16
h = 660/50.256
h = 13.13 cm

Find the radius of a cylinder with the same height and volume as a cube of sides 8 ft.

Solution:

Given,
Height of cube = height of cylinder = 8 ft, and
Volume of the cube = volume of cylinder
8 × 8 × 8 = 512 ft3
As we know,
Volume (V) = πr2h, here π = 3.141, h = 8 ft
=> 512 = 3.141 × r2 × 8
=> r2 = 512/3.141 × 8
=> r2 = 512/25.128
=> r2 = 20.38
=> r = 4.5 ft
Thus the radius of a cylinder with the same height and volume as a cube of sides 8 ft is 4.5 ft

The diameter and height of a cylinder are 24 in and 14 in, respectively. Find the volume of the cylinder.

Solution:

As we know,
Radius (r) = d/2, here d = diameter
= 24/2 in
= 12 in
Now, as we know
Volume (V) = πr2h, here π = 3.141, r = 12 in, h = 12 in
= 3.141 × 12 × 12 × 14
= 6.3323 × 103 in

### Oblique Cylinder

The formula to determine the volume of an oblique cylinder is the same as that of the right circular cylinder, which is given below.

Let us solve an example to understand the concept better.

The radius and height of an oblique cylindrical water tank are 12 cm and 16 cm, respectively. Find the volume of the tank in liters.

Solution:

As we know,
Volume (V) = πr2h, here π = 3.141, r = 12 cm, h = 16 cm
= 3.141 × 12 × 12 × 16
= 7237 cm3
Since,
1 litre =1000 cm3
∴ Volume = 7.237 litres

### Elliptic Cylinder

As we know, an ellipse has 2 radii. Also, the area of an ellipse with radius ‘a’ and ‘b’ is πab. The formula to calculate the volume of an elliptic cylinder is given below:

Let us solve an example to understand the concept better.

Find the volume of an elliptic cylinder with radii 16 cm and 20 cm and a height of 40 cm.

Solution:

As we know,
Volume (V) = πabh, here π = 3.141, a = 16 cm, b = 20 cm, h = 40 cm
= 3.141 × 16 ×× 20 × 40
= 4.0204 × 104 cm3

### Right Circular Hollow Cylinder

As we know, a right circular hollow cylinder is a cylinder consisting of two right circular cylinders bounded one inside the other; its volume can be obtained by subtracting the volume of the inside cylinder from that of the outside cylinder. The formula to calculate the volume of a right circular hollow cylinder is given below:

Let us solve an example to understand the concept better.

The outer radius of a plastic pipe is 240 mm, and the inner radius is 200 mm. If the pipe’s length is 100 mm, determine the volume of material used to make the pipe.

Solution:

As we know, a pipe is an example of a hollow cylinder
Volume (V) = π(R2 – r2)h, here π = 22/7 = 3.141, R = 240 m, r = 200 mm, h = 100 mm
= 3.141(2402 – 2002) × 100
= 3.141(40000 – 25600) × 100
= 3.141 × 17600 × 100
= 5.5264 × 106 mm3