Area:  Finding the area of common shapes

Levels 5-7

 

 

The objectives of this unit are to find the area of shapes involving:

*  rectangles;

*  triangles;

*  parallelograms;

*  trapeziums.

 

The area of a shape is measured in square units, such as cm2, m2, or mm2.  It is important that all lengths on a shape are converted to the same unit before its area can be found.

 

 

 

Area of a rectangle

 


                                                Area of a rectangle = length × width

 

 

 

 

 

Example 1:

Find the area of this shape:

 

 

 

 

 

 

 

 

 

Solution:  We can split this shape up into 2 rectangles.  One way of doing this is as follows:

Text Box: Area of rectangle A = 10 × 5 = 50cm2
 
Area of rectangle B = 7 × 4 = 28cm2.
 
So, total area = 78cm2
 
Note:  It is vital in area work to show your working out clearly.
 

 

 

 

 

 

 

 

 

 

 

 

 

 


* * * * *

 

Area of a parallelogram:

 

A parallelogram has an area equal to a rectangle with the same base and height measurements:

 

 

 

 

 

 

 

 

 

Therefore:  Area of a parallelogram = base × (perpendicular) height.

 

Note:  The measurements used for the base and the height must be perpendicular (i.e. at right angles) to each other.

 

Example 2:

Find the area of this parallelogram-

 

 

 

 

 

 

 


Solution:  To find the area, we need two measurements that are perpendicular to each other.  Here we see that the base is 6.8 cm and the height is 4.3 cm.

 

So area = 6.8 × 4.3 = 29.24 cm2.

* * * * *

 

Example 3:

Work out the length of the missing side.

 

 

 

 

 

 

 

 

Solution

The measurements 12cm and 4cm are perpendicular to each other and so can be used to find the area of the parallelogram: 

                                    area = 12 × 4 = 48cm2

Another way to find the area of the parallelogram would be to multiply 8cm by the unknown height.

So

                                    8 × ? = 48

                                          ? = 48 ÷ 8

                                          ? = 6cm

 

* * * * *

 

Area of a triangle

 

A triangle has an area that is half the area of the surrounding rectangle:

 

 

 

 

 

 

 

 

 


So:       area of a triangle = ½ × base × height    or   area =

 

Example 4:

Find the area of the triangle shown:

 

 

 

 

 

 

 

 

Solution:

Area =

 

* * * * *

 

Example 5:

The triangle shown has an area of 36 cm2.  Find the height of the triangle.

 

 

 

 

 

 

 

 


Solution:

Recall that a triangle has an area that is half of the surrounding rectangle.

Therefore the area of the rectangle shown below must be 72cm2.

 

 

 

 

 

 

 

 


The rectangle has length 9cm and width ?cm.

                        So  72 = 9 × ?

                                 ? = 72 ÷ 9 = 8cm

The triangle has height 8 cm.

 

* * * * *

 

Area of a trapezium

The formula for finding the area of a trapezium is: 

                        Area =

In this formula, a and b represent the lengths of the parallel sides and h represents the height of the trapezium.

 

Instead of using the formula, some people prefer to remember the steps involved in finding the area of a trapezium:

Step 1:  Add together the parallel sides

Step 2:  Multiply by the height

Step 3:  Divide by 2.

 

 

Example 6:

Find the area of the trapezium shown below:

 

 

 

 

 

 


Solution:

Using the formula:        

So:                                                 (working out the value of the brackets)

So:                              

 

OR using the steps:

Step 1: Add together the parallels sides 13 + 17 = 30

Step 2:  Multiply by the height:              30 × 6.5 = 195

Step 3:  Divide by 2:                                         195 ÷ 2 = 97.5 cm2.

 

* * * * *

 

Example 7:

Find the area of the shape shown here:

 

 

 

 

 

 

 

 

 

 


Solution:

Area of the trapezium is           

                                                                          =

 

Area of the triangle is               

 

So total area of the shape is 152 + 96 = 248cm2

 

* * * * *

 

Important points for area work

 

1) Remember to put units on your answers.

2) Remember to show your working out.

3) Learn the formulae!