everything you need to know to get a grade c · pdf filesum of interior angles of an octagon =...

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