Mathematics

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
Angle calculations
4
There are 3 types of angles in regular polygons
Angles and polygons
Interior 180 - e angles
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
Total i 5 x 180 9000
5
Bearings
A bearing is an angle measured in a
clockwise direction from due North
A bearing should always have 3 figures.
What are these bearings ?
Here are the steps to get your answer
2360
Notice that there is a 1800 difference between
the outward journey and the return journey
560
What is the bearing of Bristol from Bath ?
What is the bearing of Bath from Bristol ?
6
Units
Learn these metric conversions
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
Perimeter
The perimeter of a shape is the distance around
its outside measured in cm, m, etc.

• Be prepared to leave answers
• to circle questions in terms of ?
• especially in the non-calculator exam

26m
31.4m
Perim D (? x D) ? 2 Perim 15 (? x 15) ?
2 Perim 15 7.5?
7.85m
4.71m
18.4m
7.85 4.71 1 1 14.56m
8
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

49m2
40m2
16m2
Area (? x r x r) ? 2 Area (? x 5 x 5) ?
2 Area 12.5?
18m2
24m2
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!
10
Volume
The volume of a 3D solid shape is the amount of
space inside it measured in cm3, m3 etc.
27m3
56m3
42m3
384.65m3
11
6
Nets and surface area
12cm2
12cm2
4cm2
2
2
12cm2
4cm2
Cuboid 2 by 2 by 6
Net of the cuboid
12cm2
Volume 2 x 2 x 6 24cm3
Total surface area 12 12 12 12 4 4
56cm2
To find the surface area of a cuboid it helps to
draw the net
Find the volume and surface area of these cuboids
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
Spotting P, A V formulae
r(? 3)
4?rl
A
P

• 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
Transformations
1. Reflection
Reflect the triangle using the line y x then
the line y - x then the line x 1
14
Transformations
Describe the rotation of A to B and C to D
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
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
Give the vector for the translation from..
Vertical translation
D
C
A
B
16
Enlarge this shape by a scale factor of 2 using
centre O
Transformations
4. Enlargement
17
Constructions
Have a look at these constructions and work out
what has been done
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
A goat is tethered to a rail AB using a rope
(with a loop on) 1.5m long
19
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
All of these internal triangles are similar to
the big triangle because of the parallel lines
20
Triangle 2
Similarity
These two triangles are similar.Calculate length y
y 17.85 ? 2.1 8.5m
15.12m
17.85m
y
7.2m
Triangle 1
21
Calculating the Hypotenuse
Pythagoras Theorem
Hyp2 a2 b2
DE2 212 452
How to spot a Pythagoras question
Be prepared to leave your answer in surd form
(most likely in the non-calculator exam)
DE2 441 2025
DE2 2466
Right angled triangle
DE 49.659
Hyp2 a2 b2
DE2 32 62
No angles involved in question
DE2 9 36
Hyp2 a2 b2
DE2 45
Calculating a shorter side
162 AC2 112
DE ?9 x ?5
256 AC2 121
256 - 121 AC2
How to spot the Hypotenuse
135 AC2
11.618 AC
22
Pythagoras Questions
Look out for the following Pythagoras questions
in disguise
23
Calculating an angle
Trigonometry
SOHCAHTOA
Tan ? O/A
How to spot a Trigonometry question
H
Tan ? 26/53
Tan ? 0.491
O
Right angled triangle
A
An angle involved in question
Calculating a side
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
24
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
Circle angle theorems
Rule 2 - Angles in the same segment are equal
Which angles are equal here?
Big fish ?!
26
Circle angle theorems
An arrowhead
A little fish
A mini quadrilateral
Look out for the angle at the centre being part
of a isosceles triangle
Three radii
27
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
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
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)
900
900