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GCSE/KS3 β Perimeter & Area
Exercise 1 β 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 324ππ2. 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 π.
Question 8
[AQA GCSE Nov 2015 2F Q10] The diagram
shows a rectangle.
The perimeter of the rectangle is 28 ππ. Work
out the area of the rectangle.
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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 π and a
square π.
The shape has a perimeter of 44 π . The area
of the square is 36 ππ2. 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 π₯.
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.
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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 Γ 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
π΄π΅πΆπ· 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
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Exercise 2 β 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 Β£35 per m2 of tarmac and Β£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
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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 β 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
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4. If the area is
50cm2 and
the base
20cm, what is
the
perpendicular height?
5. [JMC 2014 Q9]
Triangles πππ
and πππ are
drawn on a
square grid. What fraction of the area
of triangle πππ is the area of triangle
πππ ?
6. [JMC 2009 Q6] Each square in
the figure is 1 unit by 1 unit.
What is the area of triangle
π΄π΅π (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 Β£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 πππ
has a right angle at π. The
points π , π and π divide the
side ππ into quarters. Which
of the following statements about the
areas of the triangles
πππ , ππ π, πππ, πππ is true?
A All have the same area
B Ξπππ is biggest C Ξππ π is biggest
D Ξπππ is biggest E Ξπππ is biggest
10. The area of this trapezium is
51cm2. Determine π .
If π΄π΅ = 9cm, π΄πΆ = 6ππ, π· is a point
such that πΆπ· is perpendicular to π΄π΅
and πΆπ· = 4ππ, πΈ is a point such that
π΄πΆ is perpendicular to π΅πΈ, then what
is the length of π΅πΈ?
In the diagram, π΄π΅ = π΄π· = π΄πΈ
and π΅π· = 10cm. Determine the length
π΅πΈ, leaving your answer as a fraction.
7
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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 triangleβs 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 π΄, π΅, πΆ are a
quarter of the way along the
sides as shown. What is the area of the
equilateral triangle π΄π΅πΆ?
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
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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 ππΆ?
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 βB sectionβ Junior
Maths Olympiad problem, so ordinarily, when in
the actual JMO exam, youβd 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 wouldnβt be sufficient for full marks)
Question 2:
[SMC 2011 Q16] πππ π is a
rectangle. The area of triangle
ππ π is 1
5 of the area of πππ π, and
the area of triangle πππ is 1
8 of
the area of πππ π. What fraction of the
area πππ π is the area of triangle πππ?
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?
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