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9.06 Volume of prisms and cylinders

Lesson

Introduction

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 \text{(mm)}^3, cubic centimetres \text{(cm)}^3 and cubic metres \text{(m)}^3.

Volume of prisms

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.

Rectangular prism shows its height, length, and breadth.

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.

Vertical triangular prism with its height labelled.

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.

Vertical, horizontal, and right triangular prisms.

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.

Horizontal triangular prism with its height, base, and length labelled.

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

Exploration

Use the applet below to explore the volume of prisms with triangular and rectangular bases.

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The base area of a triangular prism where triangle serves as its base can be find by the area of triangle formula A=\dfrac12 \text{ base $\times$ height} and the base area of a rectangular prism can be find by the area of a rectangle formula A=\text{length $\times$ width}. To find the volume, this area is multiplied by the height of the prism.

Examples

Example 1

Find the volume of the triangular prism shown.

A triangular prism with length 11 millimetres. The base and height of the triangle are both 4 millimetres.
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 bhUse the area of a triangle formula
\displaystyle =\displaystyle \dfrac12 \times 4 \times 4Substitute b=4 and h=4
\displaystyle =\displaystyle 8 \text{ mm}^2Evaluate
\displaystyle V\displaystyle =\displaystyle AhUse the volume of a triangular prism formula
\displaystyle =\displaystyle 8 \times 11Substitute A=8 and h=11
\displaystyle =\displaystyle 88 \text{ mm}^3Evaluate

Example 2

Find the volume of the figure shown.

A trapezoidal prism with a base area of 30 centimetres squares  and a height of 8 centimetres.
Worked Solution
Create a strategy

Use the volume of a prism formula.

Apply the idea
\displaystyle V\displaystyle =\displaystyle AhUse the volume of a prism formula
\displaystyle =\displaystyle 30 \times 8Substitute A=30 and h=8
\displaystyle =\displaystyle 240 \text{ cm}^3Evaluate
Idea summary

The volume of any prism is given by:

\displaystyle V=Ah
\bm{A}
is the area of the base of the prism
\bm{h}
is the height of the prism

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.

\displaystyle \text{Volume}\displaystyle =\displaystyle \text{Area of base $\times$ Height of prism}Copy the area of a prism formula
\displaystyle =\displaystyle \pi r ^2 \times hSubstitute \text{Area of base $ = \pi r ^2$}
\displaystyle =\displaystyle \pi r ^2 hSimplify

Exploration

Use the applet below to explore the volume of cylinders.

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The height and radius of a cylinder can be used to find its volume.

Examples

Example 3

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

A cylinder with a height of 14 centimetres and a radius of 8 centimetres.
Worked Solution
Create a strategy

Use the area of a cylinder formula.

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

The volume of a cylinder is given by:

\displaystyle V = \pi r^2 h
\bm{r}
is the radius of the cylinder
\bm{h}
is the height of the cylinder

Unknown dimension of a prism

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

Example 4

A prism has a volume of 990 \text{ cm}^3.

If it has a base area of 110 \text{ cm}^2, find the height of the prism.

Worked Solution
Create a strategy

Use the volume of a prism formula, V=Ah.

Apply the idea
\displaystyle V\displaystyle =\displaystyle AhUse the formula
\displaystyle 990\displaystyle =\displaystyle 110hSubstitute A=110 and V=990
\displaystyle \dfrac{110h}{110}\displaystyle =\displaystyle \dfrac{990}{110}Divide both sides by 110
\displaystyle h\displaystyle =\displaystyle 9 \text{ cm}Evaluate
Idea summary

If we know the volume of a prism or cylinder, we can find a missing dimension by substituting the volume and given dimensions into the appropriate volume formula.

Outcomes

MA4-14MG

uses formulas to calculate the volumes of prisms and cylinders, and converts between units of volume

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