9.06 Volume of prisms and cylinders

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.

The volume of a rectangular prism is given by

\text{Volume} = \text{Length $\times$ Breadth $\times$ Height, or}\\ V = l \times b \times h

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 $=$ Area of a triangle $\times$ 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.

\text{Volume $=$ Area of a triangle $\times$ Height of the prism, or}\\ V = Ah

For all prisms:

\text{Volume $=$ Area of the base $\times$ Height of the prism, or}\\ V=Ah

Examples

Example 1

Find the volume of the triangular prism shown.

Worked Solution

Create a strategy

Find the area of the triangle, then use the volume of a triangular prism formula.

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle \dfrac12 bh	Use the area of a triangle formula
	\displaystyle =	\displaystyle \dfrac12 \times 4 \times 4	Substitute b=4 and h=4
	\displaystyle =	\displaystyle 8 \text{ mm}^2	Evaluate
\displaystyle V	\displaystyle =	\displaystyle Ah	Use the volume of a triangular prism formula
	\displaystyle =	\displaystyle 8 \times 11	Substitute A=8 and h=11
	\displaystyle =	\displaystyle 88 \text{ mm}^3	Evaluate

Watch question walkthrough

Example 2

Find the volume of the figure shown.

Worked Solution

Create a strategy

Use the volume of a prism formula.

Apply the idea

\displaystyle V	\displaystyle =	\displaystyle Ah	Use the volume of a prism formula
	\displaystyle =	\displaystyle 30 \times 8	Substitute A=30 and h=8
	\displaystyle =	\displaystyle 240 \text{ cm}^3	Evaluate

Watch question walkthrough

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.

Examples

Example 3

Find the volume of the cylinder rounded to two decimal places.

Worked Solution

Create a strategy

Use the area of a cylinder formula.

Apply the idea

\displaystyle V	\displaystyle =	\displaystyle \pi r^2 h	Use the formula
	\displaystyle =	\displaystyle \pi \times 8^2 \times 14	Substitute r=8 and h=14
	\displaystyle =	\displaystyle 2814.87 \text{ cm}^3	Evaluate and round

Watch question walkthrough

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 times will equal the given volume?".

Examples