140

13.1 You can � nd the perimeter of shapes

The perimeter of a two-dimensional (2D) shape is the total distance around the edge of the shape.

To work out the perimeter of a rectangle you can use the following formula.Perimeter of a rectangle l w l w

2l 2w

Key Points

w

l

l � the length of the rectanglew � the width of the rectangle

Work out the perimeter of the following shape.

Missing length 8 3 5 cm

Perimeter 3 6 5 3 8 9 34 cm

First, fi nd the missing length.

Then, add the lengths of all six sides.

8 cm

3 cm

6 cm

3 cm

9 cm

Exam Tip

Make sure that you know the lengths of all the sides before you work out the perimeter.

The diagram shows a � eld in the shape 50 m

100 m80 m

110 m

of a trapezium.The farmer wants to put fencing all the way round the edge of the � eld.Fencing costs £6.50 per metre.The farmer has a budget of £2000.Can he afford to buy the fencing?

Perimeter 80 50 100 110 340 m

Total cost 340 6.50 £2210

The farmer cannot aff ord to buy the fencing as it costs £2210 and he has only £2000.

First, work out the perimeter.

Use the perimeter to fi nd the total cost.

Make your fi nal decision clear.

Example 1

Example 2

13 PERIMETER AND AREA OF 2D SHAPES

13 PERIMETER AND AREA OF 2D SHAPES

141

Exercise 13A

1 Work out the perimeters of the following shapes.a b c

2 Each side of a regular pentagon has length 8 cm. 8 cmWork out the perimeter of the pentagon.

3 Work out the perimeter of this trapezium.

8 cm

3 cm12 cm

8 cm

3 cm

4 The perimeter of an equilateral triangle is 21.6 cm. Work out the length of each side.

5 The diagram shows Mrs Smith's lawn.She wants to put edging all the way round the edge of her lawn.Edging costs £3.80 per metre.Mrs Smith has a budget of £100.Can she afford the edging? Explain your answer.

8 m6 m

3 m

4 m

5.7 m

3.8 m

4.2 m

5 cm

8 cm 8 cm

5 cm

4 cm4 cm

3 cm

2 cm2 cm

Questions in this chapter are targeted at the grades indicated.

F

E

DAO3

142

13.2 You can remember and use formulae to � nd the area of triangles and quadrilaterals

Area of a rectangle length width l w

Area of a parallelogram base vertical height

b h

Area of a trapezium

1 _ 2 sum of the lengths of the parallel sides distance between the parallel sides

1 _ 2 (a b)h

Key Points

l

w

h

b

b

h

h

b

Area of a triangle 1 _ 2 base vertical height

1 _ 2 b h

Area of a square length length l l l 2

l

l

b

a

h

(a � b)12

b

h

Work out the area of this triangle.

10 cm

6 cm

Exam Tip

Always remember to include the units in your answer.

Area of triangle

1 __ 2 base vertical height

1 __ 2 b h

1 __ 2 10 6

30 cm2

Choose the formula to use.

Put in the values of b and h.

Example 3

13 PERIMETER AND AREA OF 2D SHAPES

143

Area of trapezium 1 __ 2 sum of parallel sides distance between parallel sides

1 __ 2 (a b)h

1 __ 2 (5 7) 4

1 __ 2 12 4

24 cm2

Choose the formula to use.

Put in the values of a, b and h.

Work out the area of this trapezium.

7 cm

5 cm

4 cm

Example 4

Exercise 13B

1 A decorator wants to paint two rectangular walls. The walls are 4.5 m by 2.6 m and 3.1 m by 2.6 m.What is the total area of the two walls to be painted?

2 Find the area of the following triangles.a

4 cm

3 cm

b

6 cm

5 cm

c

3 A company makes � ags in this shape.

18 cm

10 cmIt makes 50 identical � ags. Work out the area of fabric used to make these � ags.

4 A tiler creates the following pattern using

200 cm

240 cmparallelogram-shaped tiles. Work out the total area covered by the red tiles.

5 Work out the area of each trapezium.a

9 cm

2 cm

5 cm b 12 cm

5 cm

4 cm

6 A trapezium has an area of 40 cm2.

13 cm

7 cm

hThe two parallel sides have lengths 7 cm and 13 cm. The distance between the two parallel sides is h cm. Work out the value of h.

4.8 cm

9 cm

3.1 m4.5 m

2.6 mE

D

C

AO3

144

13.3 You can � nd the areas of more complicated shapes

To � nd the area of more complicated shapes you will need to split the shape into a number of simpler shapes such as rectangles, squares, triangles or parallelograms. You can then � nd the area of each part and add these areas together to � nd the total area.

Key Point

Find the area of this shape.

10 cm

5 cm

4 cm

3 cm

Area of rectangle A 3 4 12 cm2

Area of rectangle B 10 5 50 cm2

Total area 12 50 62 cm2

Work out the area of rectangle A.

Add the two areas to fi nd the total area of the shape.

Exam Tip

Remember to show how you split up the shape. You might get marks for this in the exam.

10 cm

5 cm

4 cmA

B

3 cm

Split the shape into two rectangles, A and B

Work out the area of rectangle B.

The diagram shows the plan of a garden.Mr Jones wants to spread grass seed

6 m

6 m

2 m

11 m

over the garden.One box of grass seed is enough for 10 m2.How many boxes of grass seed will he need

Area of shape area of triangle area of square

Area of square l 2

62

36 m2

Split into simpler shapes.

Calculate the area of the square.

6 m

6 m

2 m

11 m5 m

4 m

Example 5

Example 6

13 PERIMETER AND AREA OF 2D SHAPES

145

Area of triangle 1 __ 2 b h 1 __ 2 5 4 10 m2

Total area 10 36 46 m2

Amount of grass seed 46 10 4.6 5 boxes

Work out any unknown lengths you need and mark them on the diagram. Then calculate the area of the triangle.

Add the two areas to fi nd the total area of the shape.

The number has got to be the next whole number higher than the answer, not just rounded to the nearest whole number.

Exercise 13C

1 The � oor of the hall in a house is a 225 cm by 150 cm rectangle. Tiles which are squares of side 15 cm are used to tile the � oor. Work out how many tiles are needed.

2 Liam wants to replace the carpet in his room. The � oor of the room is a rectangle measuring 4 metres by 3 metres. The carpet he wants to buy costs £8.65 per square metre. Work out how much it will cost Liam to buy enough carpet to cover the � oor.

3 Libby wants to buy some grass seed so that she can sow a new lawn in her garden. She wants the lawn to be a rectangle measuring 3.2 metres by 2.5 metres. She needs 35 grams of lawn seed for every square metre of lawn. One box of lawn seed contains 250 g.a How many boxes of lawn seed will Libby need to buy?b How much lawn seed will be left over?

4 Find the area of the following shapes.

a 4 cm

10 cm

4 cm

8 cm

b

6 cm

5 cm

3 cm

c 6 m

3 m

6 m

3 m

5 Work out the shaded area in each diagram.

a 10 cm

20 cm10 cm

5 cm

b

4 cm8 cm

12 cm

6 cm

c

9 mm

3 mm

5 mm

7 mm

D

CAO2AO3

AO2AO3

AO2

AO2

AO3

146

Exam reviewSelf-assessment checklist

I can: � nd the perimeter of a 2D shape [p.140] � nd the area of a rectangle, when given the length, l, and the width, w, using the formula A l w [p.142]

� nd the area of a square, when given the length, l, using the formula A l2 [p.142] � nd the area of a triangle, when given the base, b, and vertical height, h, using the formula A 1 _ 2 b h [p.142]

� nd the area of a parallelogram, when given the base, b, and the vertical height, h, using the formula A b h [p.142]

� nd the area of a trapezium, when given the lengths of the parallel sides, a and b, and the distance between the parallel sides, h, using the formula A 1 _ 2 (a b) h [p.142]

� nd the area of more complicated shapes by splitting the complicated shape into easier shapes. [p.144]

1 Here is a rectangle.

20 cm

10 cmDiagram NOTaccurately drawna Work out the perimeter of the rectangle.

b Work out the area of the rectangle. Nov 2008

2 A carpet 60 cm wide is to be used to cover a rectangular � oor measuring 4 metres by 9 metres.Calculate the length of carpet needed.

3 Of� ce regulations say the gap between the desks should be 900 mm to allow for wheelchair users.A desk has a length of 2 m and a width of 1 m.

1 m

2 mDiagram NOTaccurately drawn

Six of these desks are arranged as shown in the diagram below.The gap between each desk is 900 mm.

width

length

Diagram NOTaccurately drawn

What is the total length and the total width of the of� ce space needed for these desks?

Exam practice

F

E

AO3

13 PERIMETER AND AREA OF 2D SHAPES

147

4 ABC is a right-angled triangle.

8 cm

7 cm

B

A

C

Diagram NOTaccurately drawn

AB 7 cm,BC 8 cm.Work out the area of the triangle.

June 2008

5 a Work out the area of this rectangle. Diagram NOTaccurately drawn

4.5 cm

2.5 cm

A square has an area of 324 cm2.b Work out the length of one side of the square. Diagram NOT

accurately drawnArea324 cm2

June 2007

6 The diagram shows Rob’s patio. All the corners are right angles. Diagram NOT

accurately drawn

12 m

2 m

7 m

6 m

3 m

2 m

The patio is made up of square paving stones each 50 cm by 50 cm.Work out how many of these paving stones are needed to tile Rob’s patio.

7 A room has four interior walls.

Diagram NOTaccurately drawn

6 m

2.5 m

2.5 m

8 m

6 m 8 m

Window

Door

1 m

2 m

3 m

1.5 m

Alesha paints the walls with emulsion paint. She does not paint the door.A 3 litre tin of emulsion paint covers 30 m2 of wall.Work out how many 3 litre tins she needs to buy. Show all your working.

D

AO3

AO3

148

8

3x � 1

x � 6 2x � 3

Diagram NOTaccurately drawn

In the diagram, all measurements are in centimetres.The lengths of the sides of the triangle are

x 6 2x 3 3x 1

Find an expression, in terms of x, for the perimeter of the triangle.Give your expression in its simplest form. June 2009

9

7 cm 4 cm

5 cm

7 cm

Diagram NOTaccurately drawn

Work out the area of the shape. Nov 2008

10 The diagram shows a rectangle inside a triangle.

3 cm

5 cm

12 cm

10 cm

Diagram NOTaccurately drawn

The triangle has a base of 12 cm and a height of 10 cm.The rectangle is 5 cm by 3 cm.Work out the area of the region shown shaded in the diagram.

Nov 2006

11 10 cm

4 cm2 cm

4 cm2 cm

4 cm2 cm

8 cm

Diagram NOTaccurately drawn

The diagram shows 3 small rectangles inside a large rectangle.The large rectangle is 10 cm by 8 cm.Each of the 3 small rectangles is 4 cm by 2 cm.Work out the area of the region shown shaded in the diagram. June 2007

C

AO2AO3

AO2

AO2

D

100% 100%

C candidatesD candidates

To get full marks on this question, write down the perimeter of the triangle in terms of x, in anunsimplified form first and then simplify yourexpression.

C candidatesD candidates

Achieving Full Marks

13 PERIMETER AND AREA OF 2D SHAPES

149

12

30 m

75 m

100 m

160 m

Diagram NOTaccurately drawn

The diagram shows the plan of a � eld.The farmer sells the � eld for £3 per square metre.Work out the total amount of money the farmer should get. March 2007

13 Mrs Kunal’s garden is in the shape of a rectangle.Part of the garden is a patio in the shape of a triangle.The rest of the garden is grass.

12 m

7 m

6 m

patio

grass

3 m

Diagram NOTaccurately drawn

Mrs Kunal wants to spread fertiliser over all her grass.One box of fertiliser is enough for 32 m2 of grass.How many boxes of fertiliser will she need?You must show your working. Nov 2010

14 The diagram shows a garden with a pond.

7 m 4 m

8 m

3 mThe pond is in the shape of a triangle.George is going to cover the shaded area with gravel.35 kg of gravel is needed to cover 1 m2.How much gravel does George need?

15 A piece of card is in the shape of a trapezium.

6 cm

Diagram NOTaccurately drawn

10 cm

7 cm

5 cm

8 cm

3 cm

A hole is cut in the card. The hole is in the shape of a trapezium.

Work out the area of the shaded region. March 2011

C

AO3

AO2

AO3