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5.07 Applications of volume

Lesson

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.

Converting units of volume

When converting between units of volume we need to multiply by the conversion factor 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

 

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.

Volume Capacity
$1$1 cm3 $1$1 mL
$1000$1000 cm3 $1$1 L
$1$1 m3 $1000$1000 L
$1$1 m3 $1$1 kL
$1000$1000 kL $1$1 ML

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.

Outcomes

1.3.3

calculate the volumes of standard three-dimensional objects, such as spheres, rectangular prisms, cylinders, cones, pyramids and composites in practical situations, for example, the volume of water contained in a swimming pool

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