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Volume of Solids

Volume of solids is the space inside the three-dimensional figures viz. solids. In other words, volume of a solid is the amount of space it takes.

Volume of solids is measured in cubic units such as mm³, cm³, m³, and so on.

Some examples of solids are a cube, rectangular prism, triangular prism, cylinder, cone, etc. shown below.

volume of solids_1

As you can see some of the solids have edges and some don’t.

Solids that have edges and have a uniform cross sections are called prisms. Cubes, rectangular prisms and triangular prisms are examples of prisms.

Cylinder is a special type of prism, and cone is a special type of pyramid. We will look at calculating the volume of cylinder and cone separately. But first, we’ll look at calculating the volume of prisms.

 

Volume of Prisms

If you sliced the prisms several times parallel to one of the faces, the resulting cross-section will all have the same size and shape. This is shown in the diagram below.

volume of solids_2

 

volume of solids_3Volume of a prism is determined by dividing the prism into small cubes of side length 1 unit, and then counting the cubes. If each small cube has a side length of 1 unit, the its volume is 1 unit³ (read as 1 cubic unit).

 

In this cube we can see that there are three cross -sections, and each layer (i.e. cross section) has been divided into 3 lines going both length-wise and breadth-wise. In other words, the area of 1 layer is made up of 9 ‘unit cubes’.

There are three such layers (each with 9 ‘unit cubes’) so the total volume of the cube is 9 + 9 + 9 = 27 unit cubes.

Now instead of counting all the cubes in a solid, we could count the number of cubes in one layer, and then multiply them by the number of layers.

So the volume of cube = 9 cubes × 3 layers = 27 cube units

This is equivalent to calculating the cross-sectional area A, and then multiplying that area by the height of the prism, h.

Hence the volume of a prism with cross-sectional area Am and perpendicular height h is given by the formula,

V = Ah

 

Let us look at some examples of volumes of solids:

volume of solids_4Example 1: A hexagonal prism shown here. The area of the hexagonal face is 30 cm² and its height is 8 cm.

So volume = Ah

= 30 × 8

= 240 cm³

 

volume of solids_5Example 2: In this trapezoidal prism, the area of the trapezium is 6.5 cm², and its height is 10 cm.

So volume V = Ah

= 6.5 × 10

= 65 cm³

 

volume of solids_6Example 3: The cross-sectional area of the shape is 20 m² and its perpendicular height is 12.2 m.

So V = Ah

= 20 × 12.2

= 244 m³

 

Let us now look in detail at specific solids:

Volume of a cube

Volume of a rectangular prism

Volume of a triangular prism

Volume of a cylinder

Volume of a pyramid

Volume of a cone

Volume of a sphere