# Volume

In mathematics, ‘Volume’ is a mathematical quantity that shows the amount of three-dimensional space occupied by an object or a closed surface. The unit of volume is in cubic units such as m3, cm3, in3 etc.

Sometimes, volume is also termed capacity. For example, the amount of water a cylindrical jar can occupy is measured by its volume. Check volume of cylinder here.

## Definition

A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder or a sphere.

Different shapes have different volumes. In 3D geometry, we have studied the various shapes and solids such as cubes, cuboids, cylinders, cones, etc., that are defined in three dimensions. For all these shapes, we are going to learn to find the volume.

## Unit of Volume

Volume of a solid is measured in cubic units. For example, if dimensions are given in meters, then the volume will be in cubic meters. This is the standard unit of volume in the International System of Units (SI). Similarly, other units of volume are cubic centimeters, cubic feet, cubic inches, etc.

### Example

If a cuboid has dimensions of 10cm x 3cm x 5cm, then find its volume.

Sol: Volume of cuboid = length x width x height

V = 10cm x 3cm x 5cm

V = 150 cubic centimeters

Hence, we can see here the unit of volume of cuboid is measured in cubic centimeters.

## Volume of liquid

Basically, the volume of a liquid is measured in liters, where 1 liter is equal to 1000 cubic centimeters.

1 liter = 1000 cubic centimeters = 0.001 cubic meters

Hence,

1 cubic meters = 1000 liters

Also, to measure the volume of a small amount of liquid, we use milliliters.

1 milliliters = 0.001 liter = 1 cubic centimeters

Volume of liquid is also commonly measured in gallons.

1 liter = 0.264172 US liquid gallon.

## Volume Formulas

The formula to calculate the volume of a solid in a three-dimensional space is to find the product of dimensions. Basically, the volume is equal to the product of the area and height of the shape.

Volume =  Base Area x Height

For shapes having flat surfaces such as cubes and cuboids, it is easy to find the volume. But for curved shapes like cones, cylinders and spheres, we have to consider the dimensions of their curved surface as well, such as radius or diameter.

Let us check the formulas for all the shapes.

## Volume of Shapes

 Name of geometrical shape Volume formula Cube V = a3, where a is the edge-length of cube Cuboid V = length x width x height Cone V = ⅓ πr²h Where r is the radius and h is the height of cone Cylinder V = πr²h, Where r is the radius and h is the height of cylinder Sphere V = 4/3 πr3, Where r is the radius of sphere

### Volume of other shapes

 Volume of Frustum πh/3 (R2+r2+Rr) Where ‘R’ and ‘r’ are radius of base and top of frustum, respectively Volume of Prism Base Area x Height Volume of Pyramid ⅓ (Area of base) (Height) Volume of Hemisphere ⅔ (πr3) Where r is the radius

## Volume Related Facts

• The cube has all its sides equal. Therefore, the volume will equal the cube of its side length.
• If the radius and height of a cone and a cylinder are the same, then the volume of a cone is equal to one-third of the volume of a cylinder.
• The formula of volume of cuboid and rectangular prism is the same.
• The volume of the prism depends on the shape of its base. For example, if the base is square, the volume will be (side2 x height).

## Solved Examples on Volume

Q.1: Find the volume of a cube if its side length is equal to 4 cm.

Solution: Given, the length of cube = 4cm

As we know,

Volume of cube = Side3

Therefore,

Volume of cube with 4cm length = 43 (cm)3

Volume = 64 cm3

Q.2: What is the volume of the cone if the radius is 2cm and the height is 5cm.

Solution: Given, radius of circular base of cone = 2cm

Height of cone = 5cm

As we know,

Volume of cone = ⅓ πr2h

Volume = ⅓ π (2)2 (5)

Volume = ⅓ x 22/7 x 4 x 5

Volume = 20.93 cu.cm.

Q.3: The volume of a cube is 512 cm3, Its surface area is?

Solution: Since, a3 = 512 = 8 x 8 x 8

⇒ a = 8 cm

∵ Surface area of cube = 6a2

=[6 x (8)2] cm2

=384 cm2

Q.4: A hemisphere has 3 cm radius. Calculate its volume?

Solution: Volume of the hemisphere = (2/3)πr3

= (2/3) x π x 33

= (2/3) x π x 27

= 18π cm3

Q.5. Calculate the volume of a triangular prism, whose triangular base has height = 6 cm, length of base = 8cm. Height of prism = 10 cm.

Solution: Area of triangular base of prism = ½ x 8 x 6 = 24 cm2

Volume of prism = 24 x 10 = 240 cm2

## Practice Questions

1. Find the volume of spherical shape with a diameter equal to 12cm.
2. A gas cylinder is flattened on top and bottom to get two parallel circular faces on either side of the curved surface. Find the volume if the radius of circular bases is 10 cm and the height of the cylinder is 50 cm.

## Worksheets

1. Calculate the volume of cuboid with following dimensions:

• 2cm x 3cm x 4cm
• 4cm x 18cm x 5cm
• 10cm x 10cm x 10cm
• 6m x 9cm x 11 cm

2. Find the volume of cube having side-length equal to:

• 3cm
• 9 inches
• 7.5 cm
• 4.4 cm
• 0.5 cm

## Frequently Asked Questions on volume

Q1

### What is volume?

Volume is a three-dimensional quantity that is used to measure the capacity of a solid shape. It means the amount of three-dimensional space a closed figure can occupy is measured by its volume.

Q2

### What is the formula for volume?

Unlike area, where two dimensions are multiplied to find the region covered by a shape, volume is measured by multiplying the area of the shape with its third dimension. For example, if the area of the rectangle is length x breadth, then the volume of the cuboid formed by elongating the dimensions of the rectangle in space is equal to:

Volume = length x breadth x height.

Q3

### What is the unit of volume?

In Maths, volume is measured in cubic units, such as cubic meters, cubic centimeters, cubic millimeters, etc.

The volume of liquid is usually measured in liters and gallons.

Q4

### Is the volume of cube and cuboid the same?

A cube has all its faces in square shape and a cuboid has rectangular faces. Therefore, the edge lengths of cube are equal to each other whereas for cuboid they are different. Hence, a cube and a cuboid cannot have the same volume.

Q5

### What is the volume of a pyramid?

The volume of a pyramid is equal to one-third of the product of area of its base and height.