1. Mathematics
Intermediate Tier
Shape and space
GCSE Revision

2. Intermediate Tier – Shape and space revision
Contents : Angle calculations
Angles and polygons
Bearings
Units
Perimeter
Area formulae
Area strategy
Volume
Nets and surface area
Spotting P, A & V formulae
Transformations
Constructions
Loci
Pythagoras Theorem
Similarity
Trigonometry
Circle angle theorems

3. Use the rules to work out all angles
“F” angles are equal
570 h i
Use the rules to work out all angles
Angle calculations a
720
210
Angles in a half turn = 1800
Angles in a triangle = 1800 j
120
350
Angles in a full turn = 3600
1620 b
1350
980
730 k l
Angles in a quadrilateral = 3600
Opposite angles are equal
1530 c d e
Angles in an isosceles triangle m 80
“Z” angles are equal
420 f g

4. There are 3 types of angles in regular polygons
Angles and polygons
Angles at =
the centre No. of sides
Exterior =
angles No. of sides
Interior = e
angles e i e c
Calculate the value of c, e and i in
regular polygons with 8, 9, 10 and
12 sides
Answers:
8 sides = 450, 450, 1350
9 sides = 400, 400, 1400
10 sides = 360, 360, 1440
12 sides = 300, 300, 1500
To calculate the total interior angles of an irregular polygon divide it up into triangles from 1 corner. Then no. of x 180
Total i
= 5 x 180
= 9000

5. Bearings N Bristol N Bath 0560 2360
A bearing should always have 3 figures.
A bearing is an
angle measured in
a clockwise
direction from due
North
What are these bearings ? N
Here are the steps to get
your answer
Bristol
2360 N
Notice that there is a 1800 difference between the outward journey and the return journey
560
Bath
0560
2360
What is the bearing of Bristol from Bath ?
What is the bearing of Bath from Bristol ?

6. Learn these rough imperial to metric conversionskm m cm mm
x 1000
x 10
x 100
÷ 1000
÷ 10
÷ 100
Length
Units
Learn these metric
conversions kl l cl ml
Capacity kg m cg mg
Weight
Imperial  Metric
5 miles  8 km
1 yard  0.9 m
12 inches  30 cm
1 inch  2.5 cm
Learn these rough imperial to metric conversions

7. Be prepared to leave answers to circle questions in terms of 
Perimeter
Be prepared to leave answers
to circle questions in terms of 
especially in the non-calculator exam
The perimeter of a shape is the distance around its outside measured in cm, m, etc.
Perimeter = 4 x L
of a square
6.5m
Circumference =  x D
of a circle 5m
26m
Perimeter = 2(L + W) of a rectangle
7.2m 2m
15cm
Perim = D + ( x D)  2
Perim = 15 + ( x 15)  2
Perim = 
31.4m 3m
Circumference =  x D
of a semi-circle
Perimeter = ? 1m
7.85m
4.71m
18.4m
= = 14.56m

8. Be prepared to leave answers to circle questions in terms of 
The area of a 2D shape is the amount of space covered by it measured in cm2, m2 etc.
Area formulae
Be prepared to leave answers
to circle questions in terms of 
especially in the non-calculator exam
Area of = L x W
square 7m
Area of = b x h
parallelogram
10m 5m 4m
Area of = (a + b) x h
Trapezium 2m 6m 5m 4m
49m2
40m2
16m2
Area of = L x W
rectangle 9m 2m
Area of = b x h
triangle 8m 9m 6m
Area of =  x r2
circle 8m
10cm
Area = ( x r x r)  2
Area = ( x 5 x 5)  2
Area = 12.5
18m2
24m2
Area of = b x h
rhombus 7m 6m 5m 3m 7m
42m2
50.24m2
7.5m2

9. Area strategy
What would you do to get the area of each of these shapes? Do them step by step! 1. 9m
1.5m 2m 8m 3. 6m 4m 2. 7m 2m
10m 6m
1.5m 5. 3m 4. 6m

10. Volume
The volume of a 3D solid shape is the amount of space inside it measured in cm3, m3 etc.
Volume of = Area at end x L
a prism 4m
A = 14m2
Volume = L x L x L
of cube 3m
27m3
56m3
Volume of = ( x r2) x L
cylinder 7m
10m 2m
Volume of = L x W x H
cuboid 3m 7m
42m3
384.65m3

11. To find the surface area of a cuboid it helps to draw the net6
Nets and surface area
12cm2
12cm2
4cm2 2 2
4cm2
12cm2
Cuboid 2 by 2 by 6
Net of the cuboid
12cm2
Volume = 2 x 2 x 6 = 24cm3
Total surface area =
= 56cm2
To find the surface area of a cuboid it helps to draw the net
Find the volume and surface area of these cuboids: 3.
5 by 5 by 5 1.
5 by 4 by 3 2.
6 by 6 by 1
V = 5 x 4 x 3 = 60cm3
V = 6 x 6 x 1 = 60cm3
V = 5 x 5 x 5 = 125cm3
SA = 94cm2
SA = 96cm2
SA = 150cm2

12. r(+ 3) 4rl r(r + l) 1d2 4 4r2 3 4r3 3 r + ½r 4l2h 1r2h 3 1rh
Spotting P, A & V formulae
r(+ 3)
4rl P A
Which of the following
expressions could be for:
Perimeter
Area
Volume
r(r + l) A
1d2 4
4r2 3
4r3 3 A A
r + ½r V
4l2h P
1r2h 3
1rh 3 V
r + 4l A V
1r 3 P
rl
3lh2
4r2h P V V A

13. y x y = x y = - x x = 1 Transformations 1. Reflection Reflect the
triangle using
the line:
y = x
then the line:
y = - x
x = 1

14. y x Transformations 2. Rotation C B A D
Describe the rotation of A to B and C to D y x
2. Rotation
When describing a rotation always state these 3 things:
No. of degrees
Direction
Centre of rotation
e.g. a rotation of 900 anti-clockwise using a centre of (0, 1) C B A D

15. 3 -4 Transformations 3. Translation Vertical translation
What happens when we translate a shape ?
The shape remains the same size and shape and the same way up – it just……
Transformations
slides
3. Translation
Horizontal translation
Use a vector
to describe a translation 3 -4
Give the vector for the translation from……..
Vertical translation D C 6 1.
A to B 6 5 2.
A to D A B -3 4 3.
B to C 4.
D to C -3 -1

16. y x Transformations 4. Enlargement O Enlarge this shape by a scale
factor of 2 using centre O
Transformations y x
4. Enlargement O

17. Have a look at these constructions and work out what has been done
Perpendicular
bisector of a line
Have a look at these constructions and work out what has been done
900
Triangle with 3
side lengths
Bisector of
an angle
600

18. Loci
A locus is a drawing of all the points which satisfy a rule or a set of constraints. Loci is just the plural of locus.
A goat is tethered to a peg in the
ground at point A using a rope 1.5m long
1.5m
Draw the locus to show
all that grass he can eat 1. A
1.5m
A goat is tethered to a rail AB using a rope
(with a loop on) 1.5m long
Draw the locus to show all that
grass he can eat 2. A B

19. Similarity Shapes are congruent if they are exactly the same shape
and exactly the same size
Similarity
Shapes are similar if they
are exactly the same shape
but different sizes
How can I
spot similar
triangles ?
These two triangles are similar
because of the parallel lines
Triangle C
Triangle B
Triangle A
All of these “internal” triangles are similar
to the big triangle because of the parallel lines

20. Triangle 2 y Triangle 1 Similarity y = 17.85  2.1 = 8.5m
These two triangles are similar.Calculate length y
y =  2.1 = 8.5m
x 2.1
Same multiplier
15.12m
17.85m
x 2.1
Multiplier =  7.2 = 2.1 y
7.2m
Triangle 1

21. ? ? ? D F E D F E A B C Pythagoras Theorem
Calculating the Hypotenuse
Pythagoras Theorem D F E
45cm
21cm ?
Calculate the size
of DE to 1 d.p.
Hyp2 = a2 + b2
How to spot a
Pythagoras
question
DE2 =
Be prepared to leave your answer
in surd form (most likely in the
non-calculator exam)
DE2 =
DE2 = 2466
DE =
Right angled
triangle D F E
6cm
3cm ?
Calculate the size
of DE in surd form
Hyp2 = a2 + b2
DE =
DE = 49.7cm
DE2 =
No angles
involved
in question
DE2 =
DE2 = 45
Hyp2 = a2 + b2
Calculating a shorter side
DE = 45
162 = AC A B C
16m
11m ?
Calculate the size
of AC to 1 d.p.
DE = 9 x 5
256 = AC
DE = 35 cm
= AC2
How to spot the
Hypotenuse
135 = AC2
135 = AC
Longest side &
opposite
= AC
AC = 11.6m

22. Look out for the following Pythagoras questions in disguise:
Finding lengths
in isosceles
triangles y x
Find the distance
between 2 co-ords O
Finding lengths
inside a circle 1
(angle in a semi
-circle = 900)
Finding lengths
inside a circle 2
(radius x 2 =
isosc triangle) O

23. ? D F E D B C Trigonometry H O Right angled A triangle An angle
Calculating an angle
Trigonometry D F E
53cm
26cm
Calculate the size
of  to 1 d.p. 
SOHCAHTOA
How to spot a
Trigonometry
question
Tan  = O/A H
Tan  = 26/53
Tan  = O
Right angled
triangle
 = 0 A
An angle
involved
in question
Calculating a side D B C
11m ?
Calculate the size
of BC to 1 d.p.
730
SOHCAHTOA
Sin  = O/H O A
Sin 73 = 11/H
Label sides H, O, A
Write SOHCAHTOA
Write out correct rule
Substitute values in
If calculating angle use
2nd func. key
H = 11/Sin 73 H
H = m

24. A F B C E D c Circle angle theorems Rule 1 - Any angle in a
semi-circle is 900 A F
Which angles
are equal
to 900 ? c B C E D

25. Which angles are equal here?
Circle angle theorems
Rule 2 - Angles in the same segment
are equal
Which angles are equal here?
Big fish ?*!

26. Circle angle theorems Rule 3 - The angle at the centre
is twice the angle at the
circumference c c c
An arrowhead
A little fish
A mini quadrilateral c c
Look out for the
angle at the centre
being part of a
isosceles triangle
Three radii

27. D A + C = 1800 C A B B + D = 1800 Circle angle theorems
Rule 4 - Opposite angles in a cyclic
quadrilateral add up to 1800 D
A + C = 1800 C A
and B
B + D = 1800

28. c Circle angle theorems Rule 5 - The angle between the tangent
and the radius is 900 c
A tangent is a line which
rests on the outside of the
circle and touches it at one
point only

29. c Circle angle theorems Rule 7 - Tangents from an external point
are equal (this usually creates a
kite with two 900 angles in…..
…… or two isosceles triangles) c
900
900