10.17 Volume of prisms, cylinders and spheres

Volume is the amount of space an objects takes up, this can be the amount of space a 3D shape occupies or the space that a substance (solid, liquid or gas) fills. It is measured using units such as cubic millimetres (mm³), cubic centimetres (cm³) and cubic metres (m³).

To find the volume of  rectangular prisms  we can multiply the three dimensions together. Multiplying the length by the breadth gives us the area of the base, we can then multiply this by the height to find the volume.

 

Triangular prisms

To find the volume of a triangular prism, we can do as we did for the rectangular prism and find the number of squares that would cover the base (area of the base) multiplied by the height.

So the volume is:

$\text{Volume }=\text{Area of the triangle }\times\text{Height of the prism }$Volume =Area of the triangle ×Height of the prism

Since the prism can look quite different depending on the triangular face and which way it is orientated we need to be cautious about which measurements we use in our calculations.

To calculate the volume of a triangular prism we need the base and perpendicular height of the triangular face as well as the length of the prism - the distance between the two triangular faces, which is also referred to as the perpendicular height of the prism

Given these three measurements the volume of a triangular prism can be found as follows.

 

Worked example

Find the volume of the triangular prism shown.

Think: We want to identify the base and perpendicular height of the triangular face and the length of the prism. We can then use the formula: $\text{Volume }=\text{Area of triangle}\times\text{height of the prism}$Volume =Area of triangle×height of the prism.

Do:

$V$V	$=$=	$\text{Area of triangle}\times\text{height of the prism}$Area of triangle×height of the prism	Write down formula.
	$=$=	$\left(\frac{1}{2}\times4\times3\right)\times6$(12​×4×3)×6	Substitute in given lengths.
	$=$=	$6\times6$6×6	Evaluate the area first
	$=$=	$36$36	Evaluate the product to find the volume

 

So the volume of the triangular prism is $36$36 cm².

Reflect: We can see that in the case of both the rectangular prism, and the triangular prism, we first calculated the area of the flat base that makes up the shape, and then multiplied this by the height of the prism. There is nothing special about these two shapes, and in fact, we can find the volume of any prism using the same method. That is, if we know the area of the base and the height of the prism we can find the volume. 

Volume of prisms

Practice questions

Question 1

Question 2

 

Cylinders

A cylinder is very similar to a prism (except for the lateral face), so the volume can be found using the same concept we have already learnt.

Practice questions

Question 3

Question 4

 

Finding an unknown side given the volume

Sometimes we might know the volume of a prism but we are missing one measurement such as the the length or height. Using division, we can work backwards from the formula to find out the missing value.

For example: since a cube has equal length, width and height, if we know its volume then we can work out its side length by taking the cube root of the volume or asking ourselves "What value multiplied by itself $3$3 times will equal the given volume?".

 

Worked example

Example 

A rectangular prism has a volume of $330$330 cm³, and has a length of $22$22 centimetres and width of $5$5 centimetres. What is the height of the prism?

Think: The volume is length multiplied by width multiplied by height. To find an unknown value we need to divide the volume by the known values. We can write this formally in an equation and rearrange the equation to find the height.

Do:

$V$V	$=$=	$L\times W\times H$L×W×H	Write the formula.
$330$330	$=$=	$22\times5\times H$22×5×H	Substitute in known values.
$330$330	$=$=	$110H$110H	Simplify the right hand side.
$\frac{330}{110}$330110​	$=$=	$\frac{110H}{110}$110H110​	Divide both sides by $110$110.
$H$H	$=$=	$3$3	Simplify the quotients

 

We can see that the prism has a height of $3$3 cm.

Practice question

Question 5

 

 

Volume of a sphere

A sphere is defined as the collection of points that are all equal distance from the centre of the sphere. The distance from the centre is the radius of the sphere.

The volume of a sphere with radius $r$r can be calculated using the following formula.

 

Practice questions

question 6

question 7