AustraliaNSW
Stage 5.1-3

# 7.09 Further composite solids

Lesson
Summary

For two similar figures with a scale factor of $s$s, the volume of the larger figure will be equal to $s^3$s3 times greater than the volume of the smaller figure.

The reason for this is that volume is calculated using three dimensions.

### Volumes of truncated pyramids and cones

Truncated pyramids and cones are made from the difference between similar figures, so we can calculate their volumes using the scale factor between the solids.

#### Worked example

A frustum is made by truncating the top half of a pyramid.

If the original pyramid had a volume of $112$112, what is the volume of the frustum?

Think: Since the original pyramid was truncated to half its height, we know that the scale factor between the removed section and the original pyramid will be $2$2. Using this, we can calculate what fraction of the original pyramid was removed and then find the volume of the frustum.

Do: Since the scale factor is $2$2, we know that the original pyramid is $2^3$23 times greater in volume than the removed section. In other words, the removed section had a volume equal to $\frac{1}{8}$18 of the original pyramid's volume.

This means that the frustum's volume is equal to $\frac{7}{8}$78 of the original pyramid, so it will have a volume of:

$\text{Volume of the frustum}=112\times\frac{7}{8}$Volume of the frustum=112×78 $=$= $98$98

As such, knowing the scale factor between similar figures can be a quick and easy way to find the volume of composite shapes that are formed from them.

#### Practice questions

##### Question 1

A small square pyramid has a height of $x$x m and a base side length of $y$y m. A large pyramid has dimensions triple that of the small pyramid.

1. What are the dimensions of the large pyramid?

 Height $=$= $\editable{}$ m Base side length $=$= $\editable{}$ m
2. What is the volume of the large pyramid?

3. How many times can the volume of the small pyramid go into the volume of the large pyramid?

4. If the small pyramid has a volume of $46$46 m3, what is the volume of the large pyramid?

##### Question 2

Lucy makes a truncated cone by cutting off a smaller cone halfway from the top, as shown in the diagram below:

1. Find the exact volume of the original cone.

2. Find the exact volume of the cone section that was cut from the original cone.

3. What fraction of the original cone did Lucy cut off?

4. If the original cone had a volume of $184$184 cm3, what is the exact volume of Lucy's truncated cone?

##### Question 3

Rochelle has a red traffic cone and an orange traffic cone. The red traffic cone has dimensions four times that of the orange cone.

1. How many times greater is the volume of the red traffic cone compared to the orange one?

$16$16

A

$4$4

B

$12$12

C

$64$64

D

$16$16

A

$4$4

B

$12$12

C

$64$64

D
2. If the volume of the red cone is $20480$20480 cm3, what is the volume of the orange cone?

##### Question 4

A solid is constructed from two identical frustums. Each frustum was made by removing the top third of the original square pyramid. Find the volume of the solid.

1. Round your answer to the nearest two decimal places.

### Outcomes

#### MA5.3-13MG

applies formulas to find the surface areas of right pyramids, right cones, spheres and related composite solids

#### MA5.3-14MG

applies formulas to find the volumes of right pyramids, right cones, spheres and related composite solids

#### MA5.3-16MG

proves triangles are similar, and uses formal geometric reasoning to establish properties of triangles and quadrilaterals