# 5.04 Volume of cylinders and cones

Lesson

In this section we look at the volumes of cylinders and cones.

## Cylinders

We have already seen how the volume of prisms can be calculated using the rule:

$\text{Volume }=\text{area of base }\times\text{height }$Volume =area of base ×height

The base of a prism is the face that is extended, or extruded, throughout the height. Another way of identifying the base is finding the two faces that are identical and parallel to one another.

A cylinder is very similar to a prism (except for the circular face), but the volume can be found using the same process. We can think of the circle as the face that is extended throughout its height!

Volume of a cylinder
 Volume of a cylinder $=$= $\text{area of base }\times\text{height }$area of base ×height $V$V $=$= $\pi r^2\times h$πr2×h $V$V $=$= $\pi r^2h$πr2h

#### Practice question

##### Question 1

Find the volume of a cylinder correct to 1 decimal place if its radius is $6$6 cm and its height is $15$15 cm.

### Volumes formed from part of a cylinder

When we are asked to find the volume of half or quarter of a cylinder we simply multiply the full volume by the fraction we are trying to find.

#### Worked example

A solid is formed by extruding a sector, as shown in the diagram below. The sector has a radius of $4$4 m and is extruded to a height of $10$10 m. Find the volume of the solid, to the nearest cm2.

Think: The shaded sector has an angle at the centre of $78^\circ$78°. When the sector is extruded it forms a fraction of cylinder given by $\frac{78}{360}$78360.

Do: we can do our calculation using this fraction of the whole volume, that is

 Volume of solid $=$= $\frac{78}{360}\times\text{volume of cylinder }$78360​×volume of cylinder $V$V $=$= $\frac{78}{360}\times\pi r^2h$78360​×πr2h $=$= $\frac{78}{360}\times\pi\times4^2\times10$78360​×π×42×10 $=$= $108.9085$108.9085... $=$= $109$109 cm2 (nearest cm2)

#### Practice questions

##### QUESTION 2

Calculate the volume of the solid. Assume that the solid is a quarter of a cylinder.

##### QUESTION 3

Calculate the volume of the solid correct to two decimal places.

## Cones

There are many situations in which we might want to know the volume of a cone. We could work out the amount of snow cone ice that would fit inside the cone below! Or we might calculate the amount of liquid that would fit a funnel for a chemistry experiment, and then even work out the flow rate of the liquid leaving the funnel.

The volume of a cone has the same relationship to a cylinder, as a pyramid has to a prism. That is:

Volume of a cone
 Volume of a cone $=$= $\frac{1}{3}\times\text{area of base }\times\text{height }$13​×area of base ×height $V$V $=$= $\frac{1}{3}\times A\times h$13​×A×h $V$V $=$= $\frac{1}{3}Ah$13​Ah

Since the base area of the cone is a circle,

 $V$V $=$= $\frac{1}{3}\pi r^2h$13​πr2h

To calculate the volume of a cone, we must use the perpendicular height of the cone, that is, the height perpendicular to the base that meets the cone at its apex. If we are given the slant height, we can use Pythagoras' theorem with the radius to find the perpendicular height.

#### Practice questions

##### QUESTIOn 4

Find the volume of the cone shown.

##### QUESTIOn 5

Find the volume of the cone shown.

##### QUESTIOn 6

A cone has a volume of $938.29$938.29 mm3 and a height of $14$14 mm. Find the radius of the cone.

Enter each line of working as an equation.

### Outcomes

#### MS11-4

performs calculations in relation to two-dimensional and three-dimensional figures