Whenever we are using different measurements in a calculation, we need to make sure that all our measurements have the appropriate units.
In the case of calculating volume, all of our measurements need to have the same units, otherwise the numbers that we are using will be using different scales, and our calculations will be incorrect.
When finding the volume of a solid using its dimensions, we need to first choose a common unit of measurement that all the dimensions will be in terms of. Once all the dimensions have been converted to this common unit, we can then calculate the volume as per usual.
A cylindrical candle has a radius of 46 mm and a height of 0.13 m. What is the volume of the candle in cubic centimetres? Round your answer to two decimal places
The measurements must all be converted into the same unit before we can determine the volume of an object.
When converting units of volume, we can think of it as converting the units of each dimension that multiply to make up that volume. Since volume is calculated using three dimensions, any conversion between units will need to be applied three times.
Consider a cube with a side length of 1 cm.
Since there are 10 mm in each 1 cm we know that ten cubes of side length 1 mm will fit along the length, base and height of the cube. In other words, one cubic centimetre has the same volume as 10 \times 10 \times 10 cubic millimetres.
1\text{ cm}^{3} = 1000 \text{ mm}^{3}
Now consider a cube with a side length of 1 m. Since 1 m is equal to 100 cm, we can fit a hundred cubic centimetres along each dimension of the cubic metre. So one cubic metre has the same volume as 100\times 100 \times 100 cubic centimetres.
1\text{ m}^{3} = 1\,000\,000 \text{ cm}^{3}
Now what if we combined these two conversion equations?
If there are 1000 cubic millimetres in a cubic centimetre, and there are 1\,000\,000 cubic centimetres in a cubic metre, then we get:
1\text{ m}^{3} = 1\,000\,000\, 000 \text{ mm}^{3}
Volume conversions: \begin{aligned} 1\text{ cm}^{3} &= 1000\text{ mm}^{3}\\1\text{ m}^{3} &= 1\,000\,000\text{ cm}^{3}\\1\text{ m}^{3} &= 1\,000\,000\,000\text{ mm}^{3}\end{aligned}
To convert from a larger unit to a smaller unit, we multiply by the conversion factor. To convert from a smaller unit to a larger unit, we divide by the conversion factor.
Complete the working below to convert 4\text{ m}^{3} to a volume in \text{ mm}^{3}.
\displaystyle 4\text{ m}^{3} | \displaystyle = | \displaystyle 4 \times 1\,000\,000 \text{ cm}^{3} | Multiply 4 by 1\,000\,000 \text{ cm}^{3} |
\displaystyle = | \displaystyle ⬚\text{ cm}^{3} | Evaluate | |
\displaystyle = | \displaystyle ⬚ \times 1000 \text{ mm}^{3} | Multiply the number of \text{cm}^{3} by 1000\text{ mm}^{3} | |
\displaystyle = | \displaystyle ⬚\text{ mm}^{3} | Evaluate |
The outline of a trapezium-shaped block of land is pictured below.
Find the area of the block of land in square metres.
During a heavy storm, 41 mm of rain fell over the block of land. What volume of water landed on the property in litres?
Volume conversions: \begin{aligned} 1\text{ m}^{3} &= 1\,000\,000\text{ cm}^{3}\\1\text{ m}^{3} &= 1\,000\,000\text{ cm}^{3}\\1\text{ m}^{3} &= 1\,000\,000\,000\text{ cm}^{3} \\ 1\text{ m}^{3} &= 1000\text{ L}\end{aligned}
To convert from a larger unit to a smaller unit, we multiply by the conversion factor. To convert from a smaller unit to a larger unit, we divide by the conversion factor.
Similar to the way that we can convert between units of volume, we can also scale volume by applying the scale factor three times. To scale area we apply the scale factor two times.
Jeremy has an old box with dimensions of 10\text{ cm}, 12\text{ cm} and 4\text{ cm}, which has a volume of 480\text{ cm}^{3}. If Jeremy makes a new box with dimensions double that of his old box, what will its volume be?
We can scale volume by applying the scale factor three times. Or by multiplying the original volume by the scale factor cubed.