8.07 Volume of prisms and pyramids
Volume of a prism
The solid objects below are examples of prisms.
In some cases, it makes sense to refer to the 'length' of a prism, instead of the 'height; also, 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 from one end of the prism to the other.
The volume of any prism is measured in cubic units and is given by:
| $\text{Volume of a prism }$Volume of a prism | $=$= | $\text{area of end }\times\text{height }$area of end ×height |
If we wish to determine the capacity of an object, we would first calculate the volume in cubic units, and then convert the volume to appropriate units of capacity - mL, L, kL or ML.
Collecting rainwater
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 many places.
Let's have a look at an example that demonstrates how much water could be collected from the roof of a barn in an area of NSW that is often gripped by drought.
Worked example
For the barn shown:
Calculate the total rainfall in litres that could potentially 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 annual rainfall for Goulburn for the last six years has varied between $388.6$388.6 mm and $732.6$732.6 mm.
To maximise the amount of rainwater collected, we will base our calculations on the maximum rainfall for the last six years. We will also assume that the tanks need to be able to hold the whole year's 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 rectangular area that we would see if we were above the roof and looking down (regardless of any slope of the roof).
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 |
| $=$= | $31.2$31.2 m2 |
For this barn, the catchment area is a rectangle and we assumed that the maximum rainfall is $732.6$732.6 mm annually. So the volume of collected water can be considered to be a rectangular prism with a 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 collected | $=$= | $31.2\times0.7326$31.2×0.7326 |
| $=$= | $22.9$22.9 m3 | |
| Capacity of tank required | $=$= | $22.9\times1000$22.9×1000 L |
| $=$= | $22900$22900 L |
Practice questions
QUESTION 1
Find the volume of the triangular prism shown.
Question 2
A rectangular swimming pool has a length of $27$27m, width of $14$14m and depth of $3$3m.
Find the capacity of the swimming pool in Litres.
$1134$1134 L
A$113400$113400 L
B$37800$37800 L
C$1134000$1134000 L
D
Question 3
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, $63$63 mm of rain fell over the block of land.
What volume of water landed on the property in litres?
Question 4
A box of tissues is in the shape of a rectangular prism. It measures $19$19 cm by $39$39 cm by $11$11 cm.
What is the volume of the box?
A supermarket owner places tissues boxes on a shelf so that there are no gaps between the boxes or at the ends of the shelf.
If the shelf is $95$95 cm long, how many tissues boxes can be organised in this way?
(Make sure the orientation of the boxes leaves no gaps.)
Volume of a pyramid
A pyramid is formed when the vertices of a polygon are projected up to a common point (called a vertex). The polygon base of a pyramid will most commonly be a square but can be rectangular, triangular or any other polygon.
The volume of a pyramid with a base area, $A$A, and height, $h$h, is given by the formula $V=\frac{1}{3}Ah$V=13Ah
Note that the height $h$h must be measured perpendicular to the base (not along a sloped face).
For the rectangular base pyramid shown the area of the base can be calculated as $A=\text{width }\times\text{length }$A=width ×length therefore the formula for the volume is $V=\frac{1}{3}\times\text{width }\times\text{length }\times height$V=13×width ×length ×height
| $\text{Volume of pyramid}$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 pyramid must be perpendicular to the base when using this formula.
Practice questions
QUESTION 5
Find the volume of the following rectangular pyramid.
Give your answer in cubic centimetres.
QUESTION 6
Find the volume of the following pyramid.
Give your answer in cubic centimetres.
Question 7
Find the volume of a pentagonal pyramid with base area of $90$90 cm2 and height $8$8 cm.