# 9.03 Composite solids

Lesson

Composite solids are 3D figures comprising multiple simpler solids, either added together or subtracted from one another.

### Volumes of composite solids

Some composite solids are made by joining simple solids together. We can find the volume of these composite solids by finding the volume of each of the pieces.

#### Worked example

Find the volume of this composite solid, rounded to two decimal places:

Think: We can see that this composite solid is made up of a rectangular prism and a cylinder. We will find the volume of each piece separately.

Do: The volume of a rectangular prism is given by the formula $V=lbh$V=lbh, which gives us:

$\text{Volume of the rectangular prism}=7\times5\times4$Volume of the rectangular prism=7×5×4

The volume of a cylinder is given by the formula $V=\pi r^2h$V=πr2h, which gives us:

$\text{Volume of the cylinder}=\pi\times3^2\times3$Volume of the cylinder=π×32×3

Therefore, the volume of the composite solid is:

$\text{Volume }=7\times5\times4+\pi\times3^2\times3$Volume =7×5×4+π×32×3

Evaluating and rounding to two decimal places gives us a total volume of $224.82$224.82.

Reflect: Since the composite solid was made by adding simple solids together, its volume was the sum of the volumes of the simple solids.

In the case where the composite solid is made by subtracting one simple solid from another, as seen in the case below, we can subtract volumes of simple solids to find the volume of the composite solid.

This composite solid is made by subtracting half a cylinder from a rectangular prism.

In the case where the composite solid is also a prism with some composite shape as its base, we can find its volume in the same way that we'd find the volume of a prism.

### Surface areas of composite solids

The surface area of a composite solid is the sum of the areas of its faces.

In the case where the composite solid is also a prism, we can find its surface in the same way that we'd find the surface area of a prism.

If the composite solid is not a prism, we can find the surface by considering the areas of each face of the solid and adding them up.

#### Practice questions

##### Question 1

Find the volume of the composite solid shown.

##### Question 2

Find the volume of this composite solid, created by cutting a cylindrical hole through a cube, such that the cylinder's diameter is equal to the cube's side length.

##### Question 3

In the diagram, the roof has a height of $3$3 metres. Find the surface area of the figure shown.

##### Question 4

We want to find the surface area of this shape. Assume that both boxes are identical in size.

1. What is the surface area of both boxes if they had all faces exposed? Ignore where the circular faces of the cylinder touch each box for now.

2. What is the surface area of the two circular faces of the cylinder?

3. What is the area of the curved face of the cylinder?

4. Using the rounded values from the previous parts, find the surface area of the solid.