Addition and Subtraction of Volumes

Composite Solids:

Definition: A Composite Solid is a combination of basic Solids put together. Consider the diagram below, the composite solids are made as follows:

(i) A cube and a square-based pyramid

(ii) A cylinder and a cone

Hollow Shapes:

Definition: These are shapes that have space inside. The volume of material in a hollow object is found by subtracting the volume of the shape inside from the volume of the shape as if it were solid; examples of such shapes are boxes and pipes.

Frustum of Cone and Pyramid:

 Introduction: If a cone or pyramid, standing on a horizontal table is cut through, parallel to the table, the top part is a smaller cone or pyramid. The other part is called Frustum, see the diagram below.

In order to find the volume or surface area of a frustum, it is necessary to consider the frustum as a complete cone (or pyramid) with the smaller cone removed.

Example 3.4.1:

A water tank is 1.2 m square and 1.35 m deep. It is half full of water. How many times can a 9-litre bucket be filled from the tank?

Solution:

Volume = base area × height

= 1.2 × 1.2 × 1.35

= 1.944 m³

Recall, 1,000 litres = 1 m³

∴ Volume of tank in litres = 1000 × 1.944 = 1,944 litres

∴ Half Volume = \(\frac {1944}{2} \)

= 972 litres

Number of times 9 Litres will fill the half of tank

Number of times = \( \frac{972}{9} \\ \scriptsize = 108 \; times \)

Example 3.4.2: