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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. 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 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: Example 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 Example 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 Example 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