Tutor’s Toolkit: Application of Number

Carrying Out Calculations

C5	Work out areas and volumes

Useful Web Links:
BBC Bitesize Revision: Maths: Area
http://www.bbc.co.uk/schools/gcsebitesize/maths/shape/arearev1.shtml

BBC Skillswise: Numbers: Area of rectangles
http://www.bbc.co.uk/skillswise/numbers/measuring/perimeterareaandvolume/arearectangle/

C5.1	Area

Scope: 

Students need to be able to:
•	Calculate areas of rectangles and triangles.
•	Calculate areas of compound shapes composed from rectangles and triangles.
•	Interpret area units.

What students need to know:

Calculate areas of rectangles and triangles.

To calculate the area of a rectangle we multiply the length by the width. (Notice that the words breadth, width, depth and so on can be used interchangeably – we just multiply the two side lengths).

•	The units for area will always be something squared e.g. mm², cm², m² etc.
•	Make sure the 2 side lengths have the same units. If not, convert one to the units of the other.

Example Question:

Work out the area of this wooden panel:

0.8m = 80cm
320cm = 3.2m

Solution:

It is tempting to write 25,600cm²=256m² 
But this is wrong!!!
1m²=100cm  100cm=10,000cm²
So, 25,600cm²=2.56m²

We could either work out 0.8  3.2 and give the answer in m².
0.83.2=2.56m²
Or work out 80  320 and give the answer in cm².
80  320=25,600cm²

Exactly two same size triangles fit inside a rectangle with the same height and width. So to work out the area of a triangle, we work out half the area of the rectangle.

This means that we must use the height of the triangle not the length of the slanting side. This is why it is common to refer to the two dimensions as height and base.

Example Question:
Work out the area of this decoration:

Solution:
Firstly notice that the slanting side 42mm long is not relevant to the area – they just put it in to distract you!
Also, let’s work in cm. So, 38mm=3.8cm.
The area is half of 2.3  3.8.
That is 2.3  3.8  2 = 4.37cm²

Calculate areas of compound shapes composed from rectangles and triangles.

Most area calculations are actually made up of various parts, normally rectangles, sometimes with triangles included. Sometimes extra shapes are added on, sometimes they are cut out (in which case we subtract). 

Work out all of the parts separately, then add or subtract as appropriate. It often helps to add dimensions to the drawing – some of the shape’s dimensions will need to be worked out as they may only be part of a bigger dimension.

Example Question:
The diagram shows the cross section of an aircraft engine component. The shaded rectangle is a cut-out 2.5cm by 4cm. 
What is the area of the cross section?

Solution:
First let’s deal with the cut-out: 2.5  4 = 10cm². 
We’ll take that away at the end.
Now we can re-draw the diagram without the cut-out and add dimensions for two rectangles and a triangle.

The triangle is 9cm wide (12cm – 3cm) and 5cm high (the whole shape is 6 + 3 = 9cm high, so the triangle is 9cm – 4cm).
We have three shapes:
Rectangle 1: 9 x 4 = 36cm²
Rectangle 2: 3 x 3 = 9cm²
Triangle: 9 x 5  2 = 22.5cm²
Total = 36 + 9 + 22.5 = 67.5cm²

Take away the cut out (10cm)
67.5 – 10 = 57.5cm²
The area of the cross section is 57.5cm².
	
C5.2	Volume

Scope: 

Students need to be able to:
•	Calculate the volumes of cuboids.
•	Interpret packing diagrams.
•	Interpret volume units.

What students need to know:

Calculate the volumes of cuboids.
A cuboid is the mathematical name for a three dimensional shape with rectangular faces. Basically, it’s a box.

To work out the volume of a cuboid, we multiply all three dimensions together: 
e.g. length  width  depth.
•		The units for area will always be something cubed e.g. mm³, cm³, m³ etc.
•	Make sure the 3 side lengths have the same units.

For example:

Working in metres would be better to keep the size of the number down.

90cm = 0.9m
60cm = 0.6m

So, the volume is 1.4  0.9  0.6 = 0.756.
The volume is 0.76m³ (to 2 decimal places)

Be very careful comparing volume units:
1m³ = 100  100  100 = 1 000 000cm³
So, 0.76m³ = 760 000cm³

Interpret packing diagrams

When goods are packed, the outer box is made to fit a convenient number of units, without wasting any space. So, if we know the dimensions of each item and how many items go in the box, we can work out the dimensions of the box.

Example Question:
How many of the tins shown, will fit into the carton?

Solution:

Firstly, notice that the size of the tin is in mm, and the carton in cm.
150mm = 15cm and 60mm = 6cm.
15  5 = 75, so 5 tins fit along the length.
15  2 = 30, so 2 tins fit along the depth.
6  10 = 60, so 10 tins fit up the height.
In total, there will be 5  2  10 = 100 tins