# BEG CSE Shape and Space revision by hPU3zn

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```									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
“F” angles are equal             Use the
Angle calculations                               570                   rules to
Angles in a half turn = 1800                                          work out
all angles
h   i

720 a       210               Angles in a triangle = 1800

Angles in a full turn =   3600               120            j
350
b
1350
1620                     Angles in a quadrilateral = 3600
k                  730
Opposite angles are equal
980               l
1530
e d c
Angles in an isosceles triangle

“Z” angles are equal                     m
80
f        g
420
There are 3 types of angles in regular
Angles and polygons                               polygons

Angles at =       360           Exterior =   360            Interior = 180 - e
the centre       No. of sides   angles     No. of sides     angles

c
c c
e
ccc                                                e      i

Calculate the value of c, e and i in
regular polygons with 8, 9, 10 and
12 sides
8 sides = 450, 450, 1350
To calculate the total interior
9 sides = 400, 400, 1400     angles of an irregular polygon      Total i
10 sides = 360, 360, 1440   divide it up into triangles from 1   = 5 x 180
12 sides = 300, 300, 1500      corner. Then no. of x 180         = 9000
Bearings          A bearing should always            A bearing is an
have 3 figures.                    angle measured in
a clockwise
What are these bearings ?                             direction from due
North

N
Here are the steps to get
2360

N
560       Notice that there is a 1800
difference between the
outward journey and the
Bath                   return journey
0560
What is the bearing of Bristol from Bath ?                           2360
What is the bearing of Bath from Bristol ?
x 1000        x 100         x 10
Units

Learn these metric
conversions               kl
kg
km            m
l            cl
cg
cm           ml
mg
mm
Length
Weight
Capacity
÷ 1000       ÷ 100         ÷ 10

Imperial  Metric
Learn these rough             5 miles  8 km
imperial to metric            1 yard  0.9 m
conversions              12 inches  30 cm
1 inch  2.5 cm
Perimeter
The
Circumference =  x D
Perimeter = 4 x L
of a square                                           perimeter of
of a circle
Be prepared to leave answers a shape is
the distance
to circle questions in terms of around its
26m
especially in the non-calculator exam      outside
6.5m                             5m
measured
in cm, m,
Perimeter      = 2(L + W)        Perim = D + ( x D)  2 etc.
31.4m
of a rectangle
Perim = 15 + ( x 15)  2
Perim = 15 =  x D
2m                    Circumference + 7.5
15cm
of a semi-circle    2
7.85m
7.2m
Perimeter = ?
4.71m
18.4m            1m    3m       1m   = 7.85 + 4.71 + 1 + 1
= 14.56m
The area of a 2D shape is the amount of space
Area formulae              covered by it measured in cm2, m2 etc.
Area of = L x W
square              Area of       =bxh          Area of   = (a + b) x h
Beparallelogramto leave
prepared                   Trapezium
6m
49m2
to circle questions in terms of  4m
4m
40m2
5m

7m      especially in the non-calculator exam
10m
5m
2
Area of = L x W                                            2m 16m
Area of = b x h
rectangle
triangle   2
Area = ( x r x r)  2of =  x r2
Area
2m
18m2
Area = ( x 5 x circle2
9m
5) 
9m                10cm     8mArea = 12.5
Area of = b x h                  24m2
rhombus                         6m
8m
42m2                      7m                     50.24m2
6m                   3m
7.5m2
7m                                  5m
Area strategy    What would you do to get the area of each of
these shapes? Do them step by step!
1.   1.5m
4m
2.                    3.
2m
9m                                        10m
7m
2m

8m                                         6m

4.                                      3m
5.

6m                                1.5m

6m
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
Volume = L x L x L
a prism
of cube

A = 14m2
3m
4m
27m3                               56m3

Volume of = ( x r2) x L
Volume of = L x W x H
cylinder
cuboid
2m
42m3           3m          7m                 384.65m3
7m

10m
6
Nets and surface area
12cm2
12cm2         4cm2
2
2   4cm2       12cm2
Cuboid 2 by 2 by 6
Net of the cuboid
Volume = 2 x 2 x 6 = 24cm3                         12cm2

To find the surface area of a               Total surface area
cuboid it helps to draw the net              = 12 + 12 + 12 + 12 + 4 + 4
= 56cm2

Find the volume and surface area of these cuboids:
3.
1.                       2.

5 by 4 by 3               6 by 6 by 1                      5 by 5 by 5
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
Spotting P, A & V formulae
r(+ 3)                4rl
P                       A
r(r + l)
Which of the following
expressions could be for:
A
(a) Perimeter
(b) Area
1d2
(c) Volume        4r2             4      A            4r3
3     A                               3
r + ½r                       V
4l2h                                    P
V         1rh                                   1r2h
3 A               r + 4l              3
1r
V
P
3 P           4r2h      V       3lh2      V
rl A
Transformations
1. Reflection     y

Reflect the
triangle using
the line:
y=x
then the line:

x
y=-x
then the line:
x=1
Transformations      Describe the rotation of A to B and C to D

2. Rotation                            y

When describing
a rotation always
state these 3
things:

B
• No. of degrees                             C
• Direction
• Centre of
rotation
x
e.g. a rotation of
900 anti-
clockwise using
a centre of (0, 1)              A
D
What happens when we translate a shape ?
Transformations                   The shape remains the same size and shape and
the same way up – it just……. slides
.
3. Translation

Horizontal translation            Use a vector
to describe
3
a translation   -4
Vertical translation

Give the vector for
the translation
from……..
D
6
1.    A to B           0         C
6
2.    A to D
5
3.    B to C           -3
4    A             B
4.    D to C -3
-1
Enlarge this shape by a scale
Transformations           factor of 2 using centre O
4. Enlargement        y

x
O
Constructions
Perpendicular
Have a look at these    bisector of a line
constructions and work
out what has been
done

Triangle with 3
side lengths
900
Bisector of
an angle

600
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.     Draw the locus to show                                A
all that grass he can eat
1.5m

A goat is tethered to a rail AB using a rope
(with a loop on) 1.5m long                                1.5m
2.     Draw the locus to show all that               A                    B
grass he can eat
1.5m
Shapes are congruent if they
Similarity      are exactly the same shape
and exactly the same size

Shapes are similar if they                                   How can I
are exactly the same shape                                  spot similar
but different sizes                                         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
Similarity                                            Triangle 2
These two triangles are similar.Calculate length y

y = 17.85  2.1 = 8.5m
Same multiplier
x 2.1           17.85m              15.12m

y           7.2m

Triangle 1                                    x 2.1
Multiplier = 15.12  7.2 = 2.1
Calculating the Hypotenuse
Pythagoras Theorem
D                        Hyp2 = a2 + b2
How to spot a                                   DE2 = 212 + 452
Be prepared to leave your?

21cm
answer       DE2 = 441 + 2025
Pythagoras
in surd form (most likely in the
question                                      DE2 = 2466
non-calculator exam)                 DE = 2466
Right angled              F     Hyp2 = a2 + bE DE = 49.659
45cm
D
2

triangle                        the
Calculate 2 size 2    DE = 49.7cm
DE = 32 + 6
?    of DE to 1 2d.p.
3cm

DE = 9 + 36
No angles                         DE2 = 45
Calculating a shorter side     Hyp2 = a2 + b2
involved                         DE = 45              162 = AC2 + 112
F
in question       6cm         E DE = 9 x 5
A                   256 = AC2 + 121
Calculate the size
?
DE = 35 cm
of DE in surd form                     256 - 121 = AC2
How to spot the                                      135 = AC2
Hypotenuse
135 = AC
B         16m         C 11.618 = AC
Longest side &           Calculate the size
opposite              of AC to 1 d.p.               AC = 11.6m
Pythagoras Questions

Look out for the following Pythagoras questions in disguise:

y         Find the distance                  Finding lengths
in isosceles
x between 2 co-ords
triangles

x             x

Finding lengths                Finding lengths
inside a circle 1              inside a circle 2
(angle in a semi               (radius x 2 =
-circle = 900)                    isosc triangle)
O                             O
Calculating an angle
Trigonometry
D                              SOHCAHTOA
How to spot a                                           Tan  = O/A

26cm
H                  Tan  = 26/53
Trigonometry
question                                             Tan  = 0.491
O               
=   0

Right angled                   F    A 53cm             E
triangle                  Calculate the size
of  to 1 d.p.

An angle               Calculating a side
involved
in question                          D                     SOHCAHTOA
Sin  = O/H
O
?   A
Sin 73 = 11/H
•Label sides H, O, A
•Write SOHCAHTOA                                 730               H = 11/Sin 73
•Write out correct rule
•Substitute values in
B          H              C          H =         m
Calculate the size
•If calculating angle use
of BC to 1 d.p.
2nd func. key
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
Circle angle theorems

Rule 2 -     Angles in the same segment
are equal

Which angles are
equal here?

Big fish ?*!
Circle angle theorems            Rule 3 - The angle at the centre
is twice the angle at the
circumference

c
c                   c

c                                   Look out for the
c          angle at the centre
being part of a
isosceles triangle

Circle angle theorems

Rule 4 - Opposite angles in a cyclic

D
C          A + C = 1800
A
and
B
B + D = 1800
Circle angle theorems

Rule 5 -     The angle between the tangent

c

A tangent is a line which
rests on the outside of the
circle and touches it at one
point only
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

c

900

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