everything you need to know to get a grade c · pdf filesum of interior angles of an octagon =...
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EVERYTHING YOU NEED TO KNOW TO GET A GRADE C
GEOMETRY & MEASURES
(FOUNDATION)
Rhombus
Trapezium
Rectangle
Rhombus
Rhombus
Parallelogram Rhombus
Trapezium or Right-angle Trapezium
110°
Base angles in a kite are equal
Opposite angles in a rhombus are equal
250°
Angles around a point sum to 360°
Angles in a kite sum to 360°
30
Kite
Trapezium
Replace a with 3 and b with 5.2 P = 2 x 3 + 2 x 5.2
P = 6 + 10.416.4
Total areas of both shapes are equal to one as shown.
Equilateral triangle
Rhombus
2 (Fits on top of itself twice through a full turn)
9cm
5cm 5cm
5cm
3cm 3cm Can choose either as your answer
Any two of rectangle, parallelogram, kite or arrowhead
The 3cm and 5cm rods would not meet when joined with the 9cm rod.
In an isosceles triangle, the base angles are the same
Angles in a triangle sum to
180°
80 20
50 50
Each angle is 60° in an equilateral triangle
120° because angles on a straight line add to 180°
30° because angles in a right angle add to 90°
Base angles are both 30° so ABD is an isosceles triangle
Isosceles Triangle
Angles in a triangle add to 180°
180° - 34° = 146° 146° ÷ 2 = 73°
73
73°
73°
107°
107° because angles on a straight line add up to 180°
y = 180° - 107° - 38° = 35°
35
35°
No, because 38° is not equal to 35°. Therefore, it is not an isosceles triangle
180° - 126° = 54° 54° ÷ 2 = 27°
27° 27° 153°
153° because angles on a straight line add up to 180°
Angles in a triangle add to 180°
27 153
Base angles are the same in an isosceles triangle
80°
Angles in a triangle add to 180°
180° - 80° - 80° = 20°
20
Means work out angle A in triangle ABC
Means work out angle R in triangle PQR
Base angles are the same in an isosceles triangle
70°
Angles in a triangle add to 180°180° - 70° - 70° = 40°
40°
A right angle is 90°90° - 40° = 50°
50°
Both base angles are equal180° - 50° = 130° 130° ÷ 2 = 65°
65°65°
65
180° - 48° = 132° 132° ÷ 2 = 66°
66° 66°
Angles on a straight line add to 180°
180° - 66° = 114°114°
114
A quadrilateral is made up of two triangles
Angles is a triangle add up to 180°180°
180° 180° + 180° = 360°
LEARN OFF BY HEART
(because the exterior angles add up to 360°)
Exterior angle = 72°
72°
Two exterior angles joined
togetherAs worked out in part (a)
72°
144
All the angles and sides are the same in a regular pentagon
LEARN OFF BY HEART
Exterior angle
Exterior angle
Exterior angle
Exterior angle
Exterior angle
= 72°
72°
72°
72°
72°
72°
Interior angle
Angles on a straight line add up 180° Interior angle = 180° - 72° = 108°
108°
Base angles in a isosceles triangle are the same 180° - 108° = 72°
36°
36°
36
LEARN OFF BY HEARTInterior angle
Exterior angle
Angles on a straight line add up 180°
Exterior angle = 180° - 162° = 18°
= 20
20
Decagon Pentagon
Interior angle
Sum of interior angles of an decagon = (10 – 2) x 180°
Sum of interior angles = (number of sides – 2) x 180°
LEARN OFF BY HEART
= 8 x 180° = 1440°
= 144°
144°Interior angle
Sum of interior angles of a pentagon = (5 – 2) x 180°= 3 x 180° = 540°
= 108°
108°
Angles around a point add up to 360°
360° - 144° - 108°= 108°
Base angles in a isosceles triangle are the same
180° - 108° = 72° 72° ÷ 2 = 36°
144°
144° + 36° = 180° (Angles on a straight line add up to 180°) Therefore, ABC lie on a straight line
Hexagon
Square
Square
Sum of interior angles = (number of sides – 2) x 180°
LEARN OFF BY HEART
Sum of interior angles of an hexagon = (6 – 2) x 180°= 4 x 180° = 720°
= 120°
Interior angle120°
Angles around a point add up to 360°
360° - 120° - 90° - 90°= 60°
60°
Base angles are the same
180° - 60°= 120° 120° ÷ 2 = 60°
60°60°
Therefore, as all angles are 60° AHJ is equilateral
Sum of interior angles = (number of sides – 2) x 180°
LEARN OFF BY HEARTSum of interior angles of an octagon = (8 – 2) x 180° = 6 x 180° = 1080°
= 135°
135
= 135° As worked out in part (a)
135°
135°
135°
135° 135°
135°
135°
135°
Angles around a point add up to 360°
360° - 135° - 135° = 90°
Therefore, as all angles are 90° PQRS is a square
2.5 -1
Alternate angles are equal
40°
Angles on a straight line add up 180°
180° - 110° = 70°
70°
Angles in a triangle add up 180°
180° - 40° - 70° = 70°
70°
As both base angles are 70°, triangle BEF is isosceles.
55°
Alternate angles are equal
Angles in a triangle add up 180°
180° - 70° - 55° = 55°
55°
As both base angles are 55°, triangle ABC is isosceles.
41°
41
Interior angles add up to 180°
180° - 67° = 113°
113°
113
2
Perimeter of rectangle A =
Perimeter is the length around a
shape
14cmPerimeter of rectangle B = 16cm Difference = 16cm - 14cm
2
OTHER ANSWERS
ALSO ALLOWED
4cm
4cm
3cm3cm
6cm
2cm
6cm
2cm
6cm
6cm
4cm4cmPerimeter is the length around a
shape
Perimeter of rectangle = 6cm + 4cm + 6cm + 4cm
20
Square has 4 equal sides
3cm for each side
x
x
x
x
OTHER ANSWERS
ALSO ALLOWED
Because two lengths of 12cm makes 24cm which is more than the perimeter
As evident from the rectangle drawn for part (a)
40cm
1cm
40cm
1cm
20cm
2cm
20cm
2cm
10cm
4cm
10cm
4cm
8cm
5cm
8cm
5cm
Find the only rectangle which has a perimeter
of 26cm
85
Kilo means a thousand
1km = 1000m
1000m
Area = Length x Width
= 1000m x 10m
Split compound shape into two rectangles
Rec
tan
gle
A
Rectangle B
200m
Area of rectangle A = 100 x 30
Area of rectangle B = 200 x a 200a + 3000 = 10000
200a = 7000 35
Count the number of squares to find the area
C B E
Shaded Area = Area of square – Area of circle
Area of square = length x width= 80cm x 80cm
Area of circle = = 3.14 x 30cm x 30cm
Area shaded = 6400 - 28263574
LEARN THE FORMULAE OFF BY HEART
equal to (because the length around the shape is the same)
less than (because more than half the rectangle is unshaded)
Shaded Area = Area of big square ABCD – Area of the 4 congruent (identical) triangles
10cm
10
cm
Area of big square = length x width= 10cm x 10cm
Area of one triangle =
Area of one triangle =
Area of one triangle = Area of four triangles =
Shaded area =
82
Area of small square = 30cm x 30cm
900
Length of large square = 50
Area of floor = 300cm x 180cm
Number of small tiles needed =
Number of small tiles needed =
60
Perimeter is the length around a
shape
Perimeter of D = 2cm + 2cm + 2cm + 2cm
8
Perimeter A = 10cmPerimeter B = 9cm
Perimeter C = 10cm
A and C
Area is the space inside a shapeC and D
Shaded Area = Area of big square – Area of two smaller squares
Area of big square = length x width = 12cm x 12cm Area of one small square = length x width = 4cm x 4cm Area of both squares = Shaded Area =
Area of big square = length x widthArea of one small square = length x widthArea of both squares = Shaded Area =
Fraction shaded = Shaded
6
A, B and E
2 2
1
2
1
1
3
Volume is the space inside the box (number of centimetre cubes that will fit in)
8 cubes 8 more cubes required to fill box
8 cubes + 8 cubes = 16 cubes
16
Volume = length x width x height
48 = length x width x height8 x 2 x 3
Length Width
Height
8 x 74 2 x 74 3 x 108592 148 324
Round the answers
600 150 330
5cm
4cm
Volume of a cuboid = length x width x height
Volume of cuboid = 5 x 3 x 4
Volume of cube= 2 x 2 x 2
= 7.5
7
Volume of a cuboid = length x width x height
Volume of cuboid = 30 x 12 x 4
1440 3
Amount of paint = 10 x 30
93
Volume of cylinder = area of circle x length
LEARN
LEARN
Volume of cylinder = 3.14 x 3 x 3 x 10
Amount of glasses filled = 14.2 14
Cuboid A Cuboid B
Volume of cuboid A = length x width x height
Volume of cuboid A = 20 x 20 x 15
d x 20 x 20 = 7000
400d = 7000
d = 17.517.5
radius
Volume of cylinder = area of circle x length
0.5cm
x
x
x
6 RIGHT
6 DOWN
6 RIGHT
6 DOWN
6 RIGHT
6 DOWN
6 RIGHT
6 DOWN
( )6-6
180° either clockwise or anticlockwise from the origin (0,0)
2 RIGHT
4 DOWN
2 RIGHT
4 DOWN
2 RIGHT
4 DOWN
(because it’s half the size)
-2 -1
Rotation 180° either clockwise or anticlockwise from the origin (0,0)
2 RIGHT
3 DOWN
2 RIGHT
3 DOWN
2 RIGHT
3 DOWN
2 RIGHT
3 DOWN
2 RIGHT
3 DOWN
y = 1A B
C
Identical
B F
A
2 (because shape A is twice the size of shape C)
Three times bigger
= 2.4
= 2.5
The scale factor of enlargement for both respective sides must be equal.
PYTHAGORAS’ THEOREM
Hypotenuse
6.5
Angles in a triangle add up to 180°
90°
100°
A
Pythagoras’ Theorem only works in right angles triangles.
3.2
PYTHAGORAS’ THEOREM
Angles in a triangle add up to 180°
8.7
m
45°
45° angle forms an isosceles triangle. Both base length and height length of the triangle are the same.
Height of pole = 8.7 + 1.45
10.15
1m
5
Not the hypotenuse
PYTHAGORAS’ THEOREM
(a)
1.8
4.4 (Any value from 4.3 – 4.8)
£1 100p So £10 1000p
= 500
Weight of all the 2p coins = 500 x 7 = 3500g 1kg = 1000g
3.5
= 2
72 74 7678
76
= 10
310
320
330
340
350
360
370
380
390
340
= 2
82
84
86
88
87
cm or mm
litres
tonnes
LEARN
Distance = Speed x Time 1.75hours
Distance = 80 x 1.75
140
2h 15mins 135mins
Stage 2 Distance = 190 - 140 = 50km
Stage 2 Time = 2h 15mins – 1h 45mins = 30mins = 0.5hour
÷0.5 is the same as multiplying by 2
100
LEARN
45
LEARN3h 30mins 3.5hours
57.1
LEARN
Time = 5.2 hours
0.2 hour = 0.2 of 60mins
÷
x
= 12mins
5 12
NW
C
÷
x
2
N
055
Must be written as 3 figures
A
B
Scale: 1 cm represents 10 k ilom etres
037°180°
110°
290°
6.2 cm 6.2 cm x 5
31
C
026
26°
180°
115° 295°
295
L M
equidistant from two fixed points.
6.5cm
43°
Shade in the area in side the curves but out to sea (not many boats in distress on land!)
Circumference is the full length around a circle
Circumference = π x diameter
Diameter
Circumference = 3.14 x 8 Circumference = 25.12cm
Length of arc (semi-circle) = 25.12 ÷ 2 = 12.56cm
Perimeter = 12.56cm + 8cm = 20.56cm (2 d.p.)
(Total 3 marks)
Volume of prism = area of cross-section x length
Volume of prism = area of triangle x length
Volume of prism = base x height
2x length
Volume of prism = 4 x 3
2x 20
Volume of prism = 6 x 20
120