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5.03 Volume of prisms and pyramids

Lesson

Volume and units

Volume is a measure of the space inside a 3D solid shape. It is measured using units such as cubic centimetres (cm3) and cubic metres (m3).

Here is a cubic centimetre: this is a cube that is $1$1 cm long, $1$1 cm wide and $1$1 cm high.

Converting units of volume

When converting between units of volume, we need to multiply by the conversion factor for each dimension of the shape. That is, for 3D shapes we need to multiply by the conversion factor three times. For example:

If $1$1 m $=$= $100$100 cm
then $1$1 m3 $=$= $100\times100\times100$100×100×100 cm3
  $=$= $1000000$1000000 cm3

 

Capacity

The capacity of a 3D solid is the amount of material (usually liquid) that it can hold. It is measured in units such as millilitres (mL), litres (L), kilolitres (kL), and megalitres (ML).

The capacity is usually indicated on the container. For example, this carton of orange juice has a capacity of $1$1 litre.

It is useful to be able to convert between volume and capacity. The table below shows some common conversions.

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 volume and capacity of prisms and pyramids. 

 

Volume of prisms

The volume of any prism is measured in cubed units and is given by:

Volume of a prism
Volume of a prism $=$= $\text{area of end }\times\text{height }$area of end ×height

 

For a prism, the 'end' is sometimes called the 'base' or the cross-section. All prisms have a constant cross-section, meaning the cross-section remains the same size and shape across the entire 'height' of the prism.

The following applet explores how the volume of a triangular prism is affected by changes to its base and height. The dimensions of the prism can be changed by moving the sliders. Notice too, how prisms are created by pulling or extruding a face along a straight path.

 

How much water?

Area and volume become useful when dealing with rainfall and catchment. Water is a precious resource in many parts of the world and capturing rainfall from roofs and storing it in tanks is vital in a lot of areas. Let's have a look at an example that demonstrates how much water could be caught and stored on the roof of a barn in an area of NSW often gripped by drought.  

 

Worked example

For the barn shown:

Calculate the total potential rainfall in litres that could be captured from the roof and hence the size of the tank required if the barn is located in Goulburn, NSW. Assume the entire roof feeds into the storage tank.

The rainfall data for Goulburn for the last six years is shown below.

We can see from this table that the annual rainfall varied between $388.6$388.6 mm and $732.6$732.6 mm over the last six years. Let's base our calculations on the maximum rainfall for the last six years. This way we could size our tanks to maximise the amount of rain caught. We will also assume that the tanks need to be able to hold the whole years rainfall.

The first consideration is the actual catchment area of the roof? As the rain is falling from above, the actual catchment area of the roof is the area we would see if we were above the roof and looking down. The roof catchment area is sometimes called the roof footprint as shown in the image below.

Therefore, the actual catchment area of our barn roof is the same as the dimensions of the floor.

Catchment area $=$= $4.8\times6.5$4.8×6.5
Catchment area $=$= $31.2$31.2 m2

Now as the catchment area is a rectangle and we assumed out maximum rainfall is $732.6$732.6 mm annually. This forms a rectangular prism of height $732.6$732.6 mm or $0.7326$0.7326 m.

Therefore the volume of rain caught annually is equal to the volume of the rectangular prism.

Volume of rain caught $=$= $31.2\times0.7326$31.2×0.7326
Volume of rain caught $=$= $22.9$22.9 m3
Capacity of tank required $=$= $22.9\times1000$22.9×1000 L
Capacity of tank required $=$= $22900$22900 L

 

Practice questions

QUESTION 1

Find the volume of the rectangular prism shown.

QUESTION 2

Find the volume of the triangular prism shown.

Question 3

A rectangular prism has dimensions $8$8 cm by $11$11 cm by $2$2 cm. How many litres of water would it hold?

Question 4

The outline of a trapezium-shaped block of land is pictured below.

  1. Find the area of the block of land in square metres.

  2. During a heavy storm, $63$63 mm of rain fell over the block of land.

    What volume of water landed on the property in litres?

 

Volume of pyramids

The volume of a pyramid with a base area, $A$A,  and height, $h$h, is given by the formula:

Volume of a pyramid
Volume of pyramid $=$= $\frac{1}{3}\times\text{area of base }\times\text{height }$13×area of base ×height
$V$V $=$= $\frac{1}{3}Ah$13Ah

The height of the prism must be perpendicular to the base when using this formula. If we are given a slant height, we'll need to use Pythagoras' theorem to work out the vertical height for the volume calculation.

 

Practice questions

QUESTION 5

Find the volume of the rectangular pyramid shown.

QUESTION 6

We wish to find the volume of the following right pyramid.

  1. First find the vertical height.

    (Give your answer correct to 2 decimal places.)

  2. Hence find the volume to one decimal place

Question 7

A rectangular pyramid has a volume of $288$288 cm3. The base has a width of $12$12 cm and length $6$6 cm. Find the height $h$h of the pyramid.

Outcomes

MS11-4

performs calculations in relation to two-dimensional and three-dimensional figures

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