1. Parallelograms and Trapezoids
Shape and Space
The aim of this unit is to teach pupils to:
Identify and use the geometric properties of triangles, quadrilaterals and other polygons to solve problems; explain and justify inferences and deductions using mathematical reasoning
Understand congruence and similarity
Identify and use the properties of circles
Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp
Parallelograms and Trapezoids

2. Area of a parallelogram
The area of any parallelogram can be found using the formula:
Area of a parallelogram = base × perpendicular height
base
perpendicular height
Ask pupils to learn this formula.
Or using letter symbols,
Area of a parallelogram = bh

3. Area of a parallelogram
What is the area of this parallelogram?
We can ignore this length
8 cm
7 cm
12 cm
Area of a parallelogram = bh
Tell pupils that to work out the area of the parallelogram they must start by writing the formula.
They can then substitute the correct values into the formula provided that they are in the same units.
Point out that the length of the diagonal can be ignored.
Stress that it is important to always write down the correct units at the end of the calculation.
The numbers and units in the example may be modified to make the problem more challenging.
= 7 × 12
= 84 cm2

4. Area of a parallelogram
Ask pupils to suggest ways to find the area of the parallelogram. This can be done by imagining a right-angled triangle cut off one end and moved to the other. Use the pen tool set to draw straight lines to show this if required.
Drag the vertices of the parallelogram to make a rectangle. Confirm that the area of the original parallelogram and the area of the rectangle are the same.
Modify the parallelogram while keeping the length of the base and the height constant. Conclude that if the length of the base and the height of the parallelogram are kept constant then the area will remain the same regardless of the slope.
By looking at further examples deduce that the area of any parallelogram can be found by multiplying the length of the base by the perpendicular height.
Discuss the fact that we do not need to know the diagonal length on the parallelogram; warn pupils not to fall into the trap of using this measure rather than the perpendicular height.

5. The area of a parallelogram
Modify the parallelogram to change its height and its width.
If required use the pen tool to show why the area of the parallelogram is equal to the area of a rectangle with the same base length and height.
The square grid can be turned off once the derivation of the formula has been established.
As an extension activity discuss how we could use Pythagoras’ Theorem to calculate the perimeter of the parallelogram. Investigate how the perimeter changes while the area remains constant.

6. Area of a trapezium
The area of any trapezium can be found using the formula:
Area of a trapezium = (sum of parallel sides) × height 1 2
perpendicular height a b
Ask pupils to learn this formula.
Or using letter symbols,
Area of a trapezium = (a + b)h 1 2

7. What is the area of this trapezium?
Area of a trapezium
What is the area of this trapezium?
Area of a trapezium = (a + b)h 1 2
6 m
9 m
= (6 + 14) × 9 1 2
= × 20 × 9 1 2
14 m
Tell pupils that to work out the area of the trapezium they must start by writing the formula.
They can then substitute the correct values into the formula provided that they are in the same units.
Stress that it is important to always write down the correct units at the end of the calculation.
The numbers and units in the example may be modified to make the problem more challenging.
= 90 m2

8. What is the area of this trapezium?
Area of a trapezium
What is the area of this trapezium?
Area of a trapezium = (a + b)h 1 2
= (8 + 3) × 12 1 2
8 m
3 m
= × 11 × 12 1 2
12 m
This example shows a right trapezium.
Again, the numbers and units in the example may be modified to make the problem more challenging.
= 66 m2

9. Area of a trapezium
Set the pen to draw straight lines and use it to show how to divide the trapezium into triangles and rectangles. Use this to find the areas of different trapezia.
Use the pen tool to show that for any trapezium we can construct an identical trapezium rotated through 180° to make a parallelogram. If we call the lengths of the parallel sides of the trapezium a and b, then the area of the parallelogram is equal to (a + b) times the height. The area of a single trapezium is half the area of the parallelogram and so the area of any trapezium can be found using the formula ½(a + b)h.

10. The area of a trapezium
Modify the trapezium to change its height and the length of its parallel sides.
Use the pen tool to show how we could find the area of the trapezium by dissecting it into two triangles and a rectangle.
The square grid can be turned off once the derivation of the formula has been established.
As an extension activity discuss how we could use Pythagoras’ Theorem to calculate the perimeter of the trapezium.

11. What is the area of the yellow square?
Area problems
This diagram shows a yellow square inside a blue square.
3 cm
What is the area of the yellow square?
7 cm
We can work this out by subtracting the area of the four blue triangles from the area of the whole blue square.
10 cm
If the height of each blue triangle is 7 cm, then the base is
3 cm.
Talk through this area problem. Encourage the pupils to discuss strategies before you begin. There will be a temptation to try to work out or guess the lengths of the sides of the yellow square before they work out the area; ask what parts of the diagram the pupils can work out the area for instead. How does this help them answer the question?
Area of each blue triangle = ½ × 7 × 3
= ½ × 21
= 10.5 cm2

12. What is the area of the yellow square?
Area problems
This diagram shows a yellow square inside a blue square.
7 cm
10 cm
3 cm
What is the area of the yellow square?
We can work this out by subtracting the area of the four blue triangles from the area of the whole blue square.
There are four blue triangles so,
As an extension, think back to the suggestions the class made at the beginning as to how to work out the area of the square. They now know that the area of the square is 58 cm²; how can they use this knowledge to work out the length of one side of the square?
Area of four triangles = 4 × 10.5 = 42 cm2
Area of blue square = 10 × 10 = 100 cm2
Area of yellow square = 100 – 42 = 58 cm2

13. Area formulae of 2-D shapes
You should know the following formulae: b h
Area of a triangle = bh 1 2 b h
Area of a parallelogram = bh a h b
Use this slide to summarize or review key formulae.
Area of a trapezium = (a + b)h 1 2

14. Using units in formulae
Remember, when using formulae we must make sure that all values are written in the same units.
For example, find the area of this trapezium.
76 cm
Let’s write all the lengths in cm.
518 mm = 51.8 cm
518 mm
1.24 m = 124 cm
1.24 m
Stress that when substituting different lengths into a formula the units must be the same.
Link:
S7 Measures – converting units
Area of the trapezium = ½( ) × 51.8
Don’t forget to put the units at the end.
= ½ × 200 × 51.8
= 5180 cm2