Volume is a measure of the space taken up by a solid 3D object.
Volume of basic 3D shapes
In the previous lessons, we looked at how to find the surface area of prisms, cylinders, pyramids, cones, and spheres.
| 3D shape | Volume formula |
|---|---|
| Prism | $V=Ah$V=Ah |
| Cylinder | $V=\pi r^2h$V=πr2h |
| Pyramid | $V=\frac{1}{3}Ah$V=13Ah |
| Cone | $V=\frac{1}{3}\pi r^2h$V=13πr2h |
| Sphere | $V=\frac{4}{3}\pi r^3$V=43πr3 |
where $A$A is the area of the end of a shape and $h$h is the perpendicular height of a shape.
Using these formulas, we can solve real-world problems by modelling objects (or parts of objects) as 3D shapes.
Volume and units
Volume can be measured using units such as cubic centimetres (cm3) and cubic metres (m3).
A cube that is $1$1 cm long, $1$1 cm wide and $1$1 cm high has a volume of $1$1 cm3.
A cube that is $2$2 cm long, $2$2 cm wide and $2$2 cm high will fit $2\times2\times2=2^3=8$2×2×2=23=8 cubic centimetres inside it, and therefore has a volume of $8$8 cm3.
Similarly, a cube that is $100$100 cm long, $100$100 cm wide and $100$100 cm high will fit $100\times100\times100=100^3$100×100×100=1003$=$=$1000000$1000000 cubic centimetres inside it, and therefore has a volume of $1000000$1000000 cm3. Since $100$100 cm $=$= $1$1 m, we have just worked out that there are $1000000$1000000 cm3 in one cubic metre.
When converting between units of volume we need to multiply or divide by the conversion factor for lengths three times - once for each dimension of the object.
For example:
| $1$1 m | $=$= | $100$100 cm |
| therefore $1$1 m3 | $=$= | $100\times100\times100$100×100×100 cm3 |
| $=$= | $1000000$1000000 cm3 |
- Multiply if converting to a smaller unit - more smaller cubes will be needed to fill the same space
- Divide if converting to a larger unit - less larger cubes will be needed to fill the same space
- Multiply or divide by the conversion factor for lengths three times (or cubed)
Remember the conversion for lengths are:
Litres
For a 3D object which is hollow, instead of talking about the amount of space it takes up we can talk about the amount of space that fits inside, or the volume of a substance (eg. liquid or gas) that would fit inside it. This amount is called the capacity of the object. For volumes of liquids and gases the convention is to use units of litres (L) (or millilitres (mL), kilolitres (kL), and megalitres (ML) and so on). Capacity is usually (but not always) measured in litres when the object is something which typically contains liquids or gases.
Litres, millilitres, kilolitres, cubic centimetres, cubic metres, etc. are all just different units of volume (or capacity) and so we can convert between them. For example, to find the capacity of a swimming pool, it would be easier to first measure the dimensions of the pool in metres and then convert to litres. In the diagram below we can see that $1$1 cm3 is equivalent to $1$1 mL and that $1$1 m3 is equivalent to $1000$1000 L.
We may also want to convert between the following measurements:
Note: The conversion factor is $1000$1000 at each step.
In this section, we look at how to calculate the volumes and capacities of various objects.
Composite shapes
In practical problems we often encounter 3D objects that consist of two or more of the simple shapes discussed above that have been joined together in some way. These are called composite shapes. To find the volume (or capacity) of a composite shape we just add together the volumes of the simple shapes that make up the whole.
Practice questions
Question 1
This solid consists of a rectangular prism with a smaller rectangular prism cut out of it. Find the volume of the solid.
Question 2
A wedding cake with three tiers rests on a table. The layers have radii of $51$51 cm, $55$55 cm and $59$59 cm, as shown in the figure. If each layer is $20$20 cm high, calculate the total volume of the cake in cubic metres.
Round your answer to two decimal places.
