GCSE Maths: How to Calculate Volumes of a Prism
Introduction
You are bound to have met the topic of area of a two dimensional shape. A prism is a three dimensional shape and you need to be able to determine the volume. The volume of a prism is found multiplying the length, width and height together.
A prism is essentially a three dimensional shape with the same cross section i.e. the same shape going all the way through it.
For instance the following is a prism:
Prisms are three dimensional and there is a formula that can be used to determine the volume of prism given by:
Volume=Area of Cross Section × Length
In many cases the cross section consists of a compound shape so it is important that you are familiar with the area of various shapes such as that of a triangle and trapezium.
Example of Volumes Of A Prism
Take a look at the following question:
The first thing that you need to do is to identify what the cross section is i.e. what is the same shape that runs all the way through the object?
It is the “L” shape that you can see here:
The area of this shape needs to be found and it is always essential to find shapes that you can easily identify by “splitting” the shape as shown with the red line.
Now there are two rectangles A and B.
The area of rectangle A is 4 x 7 = 28 cm²
But what about the area of rectangle B? What are its dimensions?
The width is 2cm but what about the length of it? Remember it is “split” from rectangle A.
The width of rectangle A plus the length of rectangle B is 9cm and it is given that the width of rectangle A is 4cm. This means that the length of rectangle B is 5cm.
So the area of rectangle B is 5 x 2 = 10 cm²
This means that the total area of the cross section is 28 + 10 = 38 cm².
And finally using the formula to find the volume, this is:
Volume = 38 x 10 = 380 cm³
Don’t forget that you are finding a “volume” so be careful with the units.
Question Practice
Try the following volumes of a prism question on your own before looking at the solution.
Question Practice Solution
So how did you get on? Hopefully you found that the answer to be 3.5 cm
This question is a little different in that you are given the volume of the prism. What is required is to find the height of the triangle which is the cross section of the prism.
The formula for finding the volume of a prism is:
Volume=Area of Cross Section ×Length
∴84=Area of Cross Section ×12
∴Area of Cross Section=7cm²
Now the cross section is that of a right angled triangle and the formula for the area of a triangle is:
Area= \text { Area }=\frac{1}{2} \times \text { base } \times \text { height }
∴7= 12 ×4 ×Height
7=2 ×Height
∴Height=72=3.5
So the length of AB is the height of the triangle which is 3.5cm
To find the area you are in most cases, using the process for compound shapes so make sure you are determining any lengths correctly. When it comes to revising prisms for GCSE Maths, there are a number of prisms that you could be asked to determine the volume such as a cone or a cylinder. For the foundation paper your prism should either be triangular based as seen in an example above or rectangular base.