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).
For example, a cubic centimetre is a cube that is $1$1 cm long, $1$1 cm wide and $1$1 cm high.
When we find the volume of a solid shape, we are basically working out how many little cubes would fit inside the whole space. Just as when we were finding the area of a two-dimensional shape we were calculating how many unit squares would fit inside its boundary.
Exploration
If the following cube was $1$1 cubic unit, say $1$1 cm3.
Then we could calculate the volume of the following solids by counting the number of cubes. We have volumes of $2$2 cm3, $3$3 cm3 and $4$4 cm3 respectively.
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Here is a $2\times4$2×4 arrangement that is $1$1 block high. Hence, it is made up of $8$8 smaller cubes and has a volume of $8$8 cm3.
If we add another row to make the arrangement $2$2 blocks high, it will be made up of $16$16 smaller cubes and have a volume of $16$16 cm3.
Like calculating the area by dividing it into a square grid, we can calculate the volume by dividing it into an array of cubes and the number of cubes can be found by finding the number of squares that fit on the base(area of the base) by the number of rows(height). Hence, the volume of a rectangular block, is $Volume=\text{Length}\times\text{Width}\times\text{Height}$Volume=Length×Width×Height.
Rectangular prisms
Blocks like these with $6$6 rectangular faces are called rectangular prisms.
For rectangular prisms:
$Volume=\text{Length}\times\text{Width}\times\text{Height}$Volume=Length×Width×Height, this can be abbreviated to $V=L\times W\times H$V=L×W×H
A cube is a special rectangular prism where the length, width and height are all equal, so we can use the following formula:
$V=L\times L\times L$V=L×L×L, which simplifies to $V=L^3$V=L3
Note that units for volume are units cubed. Don't forget to include units for your answers!
Worked examples
Example 1
Find the volume of a cube with side length $6$6 cm.
Think: For a cube the length width and height are the same, so use the formula: $V=L^3$V=L3.
Do:
| $V$V | $=$= | $L^3$L3 | Write down formula. |
| $=$= | $(6$(6 cm$)^3$)3 | Substitute in given length. | |
| $=$= | $216$216 cm3 | Don't forget units. |
Example 2
Find the maximum volume of water the following fish tank can hold, in litres.
Think: First find the volume in cm3 using the formula for a rectangular prism. Then convert cm3 to millilitres and finally to litres using the knowledge that $1$1 cm3 is equivalent to $1$1 mL and there are $1000$1000 mL in $1$1 L.
Do:
| $V$V | $=$= | $L\times W\times H$L×W×H | Write down formula. |
| $=$= | $50\times25\times30$50×25×30 cm3 | Substitute in given lengths. | |
| $=$= | $37500$37500 cm3 | Calculate and don't forget units. |
Now convert to litres:
| $37500\ cm^3$37500 cm3 | $=$= | $37500$37500 mL |
| $=$= | $37500\div1000$37500÷1000 L | |
| $=$= | $37.5$37.5 L |
Hence, the tank has a capacity of $37.5$37.5 litres.
Practice questions
Question 1
Find the volume of the rectangular prism shown.
Question 2
A rectangular swimming pool has a length of $27$27 m, width of $14$14 m and depth of $3$3 m.
What is the volume of this swimming pool? Express your answer in m3.
Find the capacity of the swimming pool in kL.
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 we have the volume is:
$\text{Volume }=\text{Area of base }\times\text{Height }$Volume =Area of base ×Height
Which, in the case of a triangular prism, the volume can be found by:
$\text{Volume }=\text{Area of the triangle }\times\text{Height of the prism }$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 plus the 'length' of the prism - the distance between the two triangular faces.
Given these three measurements the volume of a triangular prism can be found as follows.
For triangular prisms:
$Volume=\frac{1}{2}\text{base}\times\text{height}\times\text{length}$Volume=12base×height×length
Which we can simplify to $V=\frac{1}{2}bhL$V=12bhL.
Where the base and height refer to the base and perpendicular height of the triangular face.
Worked example
Example 3
Find the volume of the triangular prism shown.
Think: Identify the base and perpendicular height of the triangular face and the length of the prism. Then so use the formula: $V=\frac{1}{2}bhL$V=12bhL.
Do:
| $V$V | $=$= | $\frac{1}{2}bhL$12bhL | Write down formula. |
| $=$= | $\frac{1}{2}\times4\times3\times6$12×4×3×6 cm3 | Substitute in given lengths. | |
| $=$= | $36$36 cm3 | Don't forget units. |
Practice questions
Question 3
Find the volume of the triangular prism shown.
Question 4
A fish tank designed to sit in the corner of a room has the shape of a right-angled triangular prism.
If its dimensions are as shown in the image, determine the capacity of the fish tank in litres.
Finding an unknown side given volume
Sometimes we might know the volume of a rectangular or triangular 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. In the case of a cube, since it has equal length, width and height, if we know its volume, 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 get the required volume?".
Worked example
Example 4
A rectangular prism has a volume of $330$330 m3, and has a length of $22$22 metres and width of $5$5 metres. 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 $330$330 by $110$110. |
| $\therefore H$∴H | $=$= | $3$3 m | Don't forget units. |
Practice questions
Question 5
Find the length, $a$a, of the rectangular prism in millimetres.
Question 6
Consider a cube with a volume of $27$27 cm3.
Complete the equation for the side length of the cube.
$s$s$=$=$\sqrt[3]{\editable{}}$3√ cm
Find the side length of the cube.
Question 7
The volume of the following tent is $4.64$4.64 m3.
Determine the height $h$h of the tent, in cm.
