gcse/ks3 perimeter & area

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GCSE/KS3 – Perimeter & Area

Exercise 1 – Rectangles/Squares

Question 1

[AQA GCSE Nov 2016 2F Q8] Work out the

perimeter of this rectangle.

Question 2

[Edexcel GCSE Nov2013-2F Q12c] Here is a

rectangle.

Work out the area of this rectangle.

Question 3

[Edexcel IGCSE May2015-2F Q8] The diagram

shows a vegetable garden in the shape of a

rectangle.

The vegetable garden has length 8.3 m and

width 4.5 m. Dan wants to put fencing

completely around the edge of the vegetable

garden. He already has 10.6 m of fencing.

How much more fencing does Dan need?

Question 4

[Edexcel GCSE June2006-2F Q20b,

June2006-4I Q2b] A square has an

area of 324𝑐𝑚2. Work out the length

of one side of the square.

Question 5

[AQA GCSE June 2014 2F Q9 Edited] Find the

dimensions of the rectangle with:

Perimeter = 20 cm and Area = 24 cm 2

Question 6

[Edexcel GCSE(9-1) Nov 2018 2F

Q13a] A square has an area of 81

cm 2

Find the perimeter of the square.

Question 7

[AQA GCSE June 2013 1F Q13b] The

perimeter of this rectangle is 20 cm.

Work out the value of 𝑙.

Question 8

[AQA GCSE Nov 2015 2F Q10] The diagram

shows a rectangle.

The perimeter of the rectangle is 28 𝑐𝑚. Work

out the area of the rectangle.


Question 9

[Edexcel GCSE(9-1) Nov 2017 1F Q8 Edited] The

length of a rectangle is twice as long as the

width of the rectangle.

The area of the rectangle is 32 cm 2. Find the

dimensions of the rectangle.

Question 10

[AQA GCSE Nov 2015 1F Q15, Nov 2015 1H Q4]

A shape is made from a rectangle 𝑅 and a

square 𝑆.

The shape has a perimeter of 44 𝑚 . The area

of the square is 36 𝑐𝑚2. Work out the area of

the shape.

Question 11

[AQA GCSE Nov 2013 1F Q14] Each small

shaded square has an area of 4 cm 2.

Work out the length 𝑥.

Question 12

[AQA GCSE Nov 2016 1F Q17, Nov 2016 1H

Q8] Field A is a rectangle with side of 30 m

and 70 m. Field B is a square with the same

perimeter as Field A.

Question 13

[AQA GCSE June 2012 1F

Q12a] The diagram

shows a rectangle.

Four of these rectangles are put together as

shown.

Work out the shaded area.

Question 14

[Edexcel IGCSE(9-1) Jan 2019 2F Q19, Jan

2019 2H Q5] Calvin has 12 identical

rectangular tiles. He arranges the tiles to fit

exactly round the edge of a shaded rectangle,

as shown in the diagram below.

Work out the area of the shaded rectangle.

Question 15

[Edexcel GCSE Nov2016-1F Q23, Nov2016-1H

Q7 Edited] The diagram shows a path around a

pond.


The pond is in the shape of a rectangle with

length 7 m and width 4 m. The path is 3 m wide.

Ali is going to cover the path with gravel.

One bag of gravel will cover 10 m 2 of the path.

How many bags of gravel does Ali need to buy?

Question 16

[Edexcel GCSE June2012-1F Q14] The

diagram shows a rectangle and a square.

The perimeter of the rectangle is the same as

the perimeter of the square. Work out the

length of one side of the square.

Question 1

[Kangaroo Grey 2015 Q3] Four identical small

rectangles are put together to form a large

rectangle as shown. The length of a shorter side

of each small rectangle is 10 cm. What is the

length of a longer side of the large rectangle?

Question 2

[Junior Kangaroo 2015 Q15] A

rectangular garden is surrounded by a

path of constant width. The

perimeter of the garden is 24 m

shorter than the distance along the outside

edge of the path. What is the width of the

path?

Question 3

Five identical

rectangles fit together

as shown. What, in

cm2, is the total area

which they cover?

Question 4

[IMC 2007 Q13] A 30cm × 40cm page of

a book includes a 2cm margin on each

side, as shown. What percentage of the

page is occupied by the margins?

Question 5

My rabbit Nibbles lives in a

moveable pen and helps to

keep the grass short. The pen is

rectangular and measures 3m

by 2m, as shown in the

diagram, where the arrow indicates North. On

successive days, the pen is moved 1m East, 2m

South, 1m West and 2m North. What is the

total area, in square metres, of the region of

grass which Nibbles can nibble?

Question 6

[IMC 2013 Q21] The square

𝐴𝐵𝐶𝐷 has an area of 196. It

contains two overlapping

squares; the larger of these

squares has an area 4 times that of the smaller

and the area of their overlap is 1. What is the

total area of the shaded regions?

Question 7

Three congruent squares overlap as

shown. The areas of the three

overlapping sections are 2 cm2, 5 cm

2 and 8

cm2 respectively. The total area of the non-

overlapping parts of the squares is 117 cm2.

What is the side-length of each square?

Question 8

[JMO 2005 B2] The diagram shows a

square which has been divided into

five congruent rectangles. The perimeter of

each rectangle is 51cm. What is the perimeter

of the square?

4


Exercise 2 – Area & Perimeter of

Rectilinear Shapes

1. Find the area and perimeter of the

following shapes:

a)

b)

2. Boris wants to build a car park. It

costs £35 per m2 of tarmac and £24

per metre of hedge to go around

the border. What is the total cost?

3. [Edexcel IGCSE(9-1) Jan 2019(R)

2F Q12]

The diagram shows the plan of the

floor in a room.

Alonso is going to cover the

floor once with polish. He buys

some tins of polish.

Each tin has enough polish to cover

14m 2 of the floor. Each tin costs

9.59 euros. Work out the total

cost of the tins that Alonso needs

to buy.

4. [Edexcel GCSE Jun2016-1F Q24,

Jun2016-1H Q7 Edited] The

diagram shows the plan of a floor.

Angie is going to varnish the floor.

She needs 1 litre of varnish for 5

m2 of floor.

There are 2.5 litres of varnish in

each tin of varnish. Angie has 3

tins of varnish.

How many tins of varnish will she

actually need? Find the area and

perimeter of the following figure.

5.

Determine the area and perimeter.

5


6. [Edexcel GCSE(9-1) Nov 2017 3F

Q24, Nov 2017 3H Q6] Here is a

rectangle.

The length of the rectangle is 7 cm

longer than the width of the

rectangle.

4 of these rectangles are used to

make this 8-sided shape.

The perimeter of the 8-sided shape

is 70 cm.

Work out the area of the 8-sided

shape.

7. [JMO 2015 A3] The diagram

shows one square inside

another. The perimeter of the

shaded region has length 24

cm. What is the area of the larger

square?

8. [Kangaroo Pink 2013 Q2] The

diagram shows six identical

squares, each containing a shaded

region. How many of the regions

have perimeter equal in length to

the perimeter of one of the

squares?

9. These

rectangles are

congruent.

Find the area

and perimeter

of the shape

Exercise 3 – Triangles,

Trapeziums and Parallelograms

1. Find the area of the following shapes.

a) b)

c) d)

e)

2. a) b)

c) d)

3. a) b)

c) d)

e)

6


4. If the area is

50cm2 and

the base

20cm, what is

the

perpendicular height?

5. [JMC 2014 Q9]

Triangles 𝑋𝑌𝑍

and 𝑃𝑄𝑅 are

drawn on a

square grid. What fraction of the area

of triangle 𝑋𝑌𝑍 is the area of triangle

𝑃𝑄𝑅?

6. [JMC 2009 Q6] Each square in

the figure is 1 unit by 1 unit.

What is the area of triangle

𝐴𝐵𝑀 (in square units)?

A 4 B 4.5 C 5

D 5.5 E 6

7. Bob has a garden in the shape of

a trapezium. If it costs £15.22

per m2 of turf, how much will it

cost to turf his garden?

8. [IMC 2005 Q9] Which of the following

shaded regions has an area different

from the other shaded regions?

9. [SMC 2003 Q2] Triangle 𝑃𝑄𝑈

has a right angle at 𝑈. The

points 𝑅, 𝑆 and 𝑇 divide the

side 𝑄𝑈 into quarters. Which

of the following statements about the

areas of the triangles

𝑃𝑄𝑅, 𝑃𝑅𝑆, 𝑃𝑆𝑇, 𝑃𝑇𝑈 is true?

A All have the same area

B Δ𝑃𝑄𝑅 is biggest C Δ𝑃𝑅𝑆 is biggest

D Δ𝑃𝑆𝑇 is biggest E Δ𝑃𝑇𝑈 is biggest

10. The area of this trapezium is

51cm2. Determine 𝑎 .

If 𝐴𝐵 = 9cm, 𝐴𝐶 = 6𝑐𝑚, 𝐷 is a point

such that 𝐶𝐷 is perpendicular to 𝐴𝐵

and 𝐶𝐷 = 4𝑐𝑚, 𝐸 is a point such that

𝐴𝐶 is perpendicular to 𝐵𝐸, then what

is the length of 𝐵𝐸?

In the diagram, 𝐴𝐵 = 𝐴𝐷 = 𝐴𝐸

and 𝐵𝐷 = 10cm. Determine the length

𝐵𝐸, leaving your answer as a fraction.

7


Exercise 4 - Fractions of Shapes

Test Your

Understanding: [JMC

2012 Q22] The

diagram shows a

design formed by

drawing six lines in a

regular hexagon. The lines divide each

edge of the hexagon into three equal parts.

What fraction of the hexagon is shaded?

A 1

5 B

2

9 C

1

4 D

3

10 E

5

16

By considering what fraction the

shaded triangle’s base and height

is of the larger triangle, work out

the fraction of the shape shaded.

Fraction of width =

Fraction of (perpendicular) height =

Overall fraction =

[SMC 2003 Q21] The outer

equilateral triangle has area 1.

The points 𝐴, 𝐵, 𝐶 are a

quarter of the way along the

sides as shown. What is the area of the

equilateral triangle 𝐴𝐵𝐶?

A 3

8 B

7

16 C

1

2 D

9

16 E

5

8

Question 1: [IMC 2011

Q18] The diagram

contains six equilateral

triangles with sides of

length 2 and a regular hexagon with sides

of length 1.

What fraction of the whole shape is

shaded?

A 1

8 B

1

7 C

1

6 D

1

5 E

1

4

Question 2: [JMC 2006

Q16] The diagram shows

an equilateral triangle

with its corners at the

mid-points of alternate sides of a regular

hexagon. What fraction of the area of the

hexagon is shaded?

A 1

2 B

1

3 C

3

8 D

4

9 E

7

12

Question 3: [JMC 2007 Q5] In

the diagram, the small squares

are all the same size. What

fraction of the large square is

shaded?

A 9

20 B

9

16 C

3

7 D

3

5 E

1

2

Question 4: [JMC 2001 Q9] In

the diagram, a corner of the

shaded star is at the midpoint

of each side of the large

square. What fraction of the

large square is covered by the star?

A 1

5 B

1

4 C

1

3 D

3

8 E

2

5

Question 5: [JMC 2015 Q22] The diagram

shows a shaded region inside a regular

hexagon. The shaded region is divided into

equilateral triangles. What fraction of the

area of the hexagon is shaded?

A 3

8 B

2

5 C

3

7 D

5

12 E

1

2

Question 6: Determine the fraction of each

square each region within it is. (e.g. the

top-left region of the square is 1

4 of the

square)

8


Question 7: [JMO 1999 A10]

What fraction of the whole

square is occupied by the

shaded square?

Question 8: [IMC 2004

Q25] The diagram shows

a square with two lines

from a corner to the

middle of an opposite

side. The rectangle fits exactly inside these

two lines and the square itself. What

fraction of the square is occupied by the

shaded rectangle?

A 1

3 B

2

5 C

3

10 D

1

2 E

3

8

Question 9: [IMC 2012 Q25]

The diagram shows a

ceramic design by the

Catalan architect Antoni

Gaudi. It is formed by

drawing eight lines

connecting points which divide the edges

of the outer regular octagon into three

equal parts, as shown.

What fraction of the octagon is shaded?

A 1

5 B

2

9 C

1

4 D

3

10 E

5

16

Question 10: [IMC

2015 Q25] A point is

marked one quarter

of the way along

each side of a triangle, as shown. What

fraction of the area of the triangle is

shaded?

A 7

16 B

1

2 C

9

16 D

5

8 E

11

16

Question 11: The diagram

shows a square ABCD of side

10 units. Line segments AP,

AQ, AR and AS divide the

square into five regions of

equal area, as shown.

What is the length of 𝑄𝐶?

Question 1: [JMO 2008 B5] In the

diagram, the rectangle ABCD is divided into

three congruent rectangles. The line

segment JK divides CDFG into two parts of

equal area. What is the area of triangle AEI

as a fraction of the area of ABCD?

(Note: This question is a ‘B section’ Junior

Maths Olympiad problem, so ordinarily, when in

the actual JMO exam, you’d be expected to

justify why you were able to break up the shape

in the way you did, using worded explanation.

Just stating the answer and showing lines in the

diagram wouldn’t be sufficient for full marks)

Question 2:

[SMC 2011 Q16] 𝑃𝑄𝑅𝑆 is a

rectangle. The area of triangle

𝑄𝑅𝑇 is 1

5 of the area of 𝑃𝑄𝑅𝑆, and

the area of triangle 𝑇𝑆𝑈 is 1

8 of

the area of 𝑃𝑄𝑅𝑆. What fraction of the

area 𝑃𝑄𝑅𝑆 is the area of triangle 𝑄𝑇𝑈?

A 27

40 B

21

40 C

1

2 D

19

40 E

23

60

Question 3:

[IMOK 2013 Solutions Back Cover] This

shape spirals inwards infinitely. What

fraction of the shape is shaded?

gcse/ks3 perimeter & area

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