2

Shape and Space Revision

- Pythagoras Theorem Slides 3 - 4

- Trigonometry Slides 5 - 8

- 2-d Shapes Slide 9

- Triangles Slide 10

- Quadrilaterals Slide 11 - 12

- Calculating Areas Slides 13 - 16

- The Circle Slides 17 - 18

- 3-d Shapes Slides 19 - 22

- Calculating Volume and Density Slides 23 - 25

- Dimensions Slides 26 - 27

- Angles Slides 28 - 33

- Transformations Slides 34 - 39

- Metric Measure Slides

3

Pythagoras’ Theorem

Pythagoras’ Theorem states :

the square of the hypotenuse of a right angled triangle is equal

to the sum of the squares of the other two sides

hypotenuse

h h2 = a2 + b2

** Notice that the hypotenuse of a right-angled triangle is the longest side and is

ALWAYS opposite the right angle.

a

b

4

Pythagoras’ Theorem

Example 1 Example 2

finding the hypotenuse finding a shorter side

h2 = a2 + b2 h2 = a2 + b2

AC2 = 182 + 212 41.52 = 32.52 + BC2

AC2 = 324 + 441 1722.25 = 1056.25 + BC2

AC2 = 765 BC2 = 1722.25 – 1056.25

AC = √765 = 27.7 cm (1d.p.) BC2 = 666

BC = √666 = 25.8cm (1d.p.)

A

B CA

B

C

18cm

21cm

41.5

cm

32.5cm

5

Trigonometry

Trigonometry is all about finding sides and angles in right-angled triangles.

There are a couple of different ways of remembering this:

1) SOH CAH TOA

2) Two Old Angles Skipped Over Heaven Carrying A Harp

adj

opptan

hyp

oppsin

hyp

adjcos

hypotenuse

adjacent

opposite

6

Trigonometry

Examples : Finding an Angle

1) 2) 3)

SOH CAH TOA SOH CAH TOA SOH CAH TOA

19.4cm

14cm

15.3cm

9.8cm

11.9cm

21.3cmhyp

hyp

hyp

opp

opp

opp

adj

adj

adj

8.35

4.19

14tan

4.19

14tan

adj

opptan

1

8.39

3.15

8.9sin

3.15

8.9sin

hyp

oppsin

1

0.56

3.21

9.11cos

3.21

9.11cos

cos

1

hypadj

7

Trigonometry

Examples : Finding a Side

1) 2) 3)

SOH CAH TOA SOH CAH TOA SOH CAH TOA

15cm

x21.5

31.3cm

xhyp

hyp

adj

adj

opp opp

adj

opp

hyp

cm0.13

41tan1515

41tan

adj

opptan

x

x

x

41°

19°

63°

x

cm

hypopp

2.19

63sin5.215.21

63sin

sin

x

x

x

m6.29

19cos3.313.31

19cos

hyp

adjcos

cx

x

x

8

Trigonometry

Examples : Finding a Side

4) 5) 6)

SOH CAH TOA SOH CAH TOA SOH CAH TOA

19.1cm

xx 4.5cm

x

hyphyp

adj

adj

adj

opp

opp

opp

hyp

cm5.15tan51

19.1

1.1951tan

adj

opptan

x

x

x

51°

73°63°

14.3cm

cmsin6314.3

hypopp

0.16

3.1463sin

sin

x

x

x

cm4.15cos73

4.5

5.473cos

hyp

adjcos

x

x

x

9

2-d Shapes

2-d Shapes are FLAT. This means that you CANNOT pick them up.

A flat shape with straight edges is known as a POLYGON.

Some polygons have been given special names :

3 sidesTriangle

4 sidesQuadrilateral

5 sidesPentagon

6 sidesHexagon

7 sidesHeptagon

8 sidesOctagon

9 sidesNonagon

10 sides Decagon

12 sides Dodecagon

10

2-d Shapes

Triangles

Equilateral Isosceles Scalene Right-Angled

- 3 equal sides - 2 equal sides - No equal sides - 1 Right Angle

- 3 equal 60° angles - 2 equal angles - No equal angles

- 3 lines of symmetry - 1 line of symmetry - No lines of Symmetry - Note that a triangle can

- Rotational Symmetry 3 - No Rotational Symmetry - No Rotational Symmetry be Right-Angled at the

same time as being

isosceles or scalene

11

2-d Shapes

Quadrilaterals

Square Rectangle Rhombus

- 4 equal sides - Opposite sides equal - 4 equal sides

- 4 right angles - 4 Right Angles - Opposite angles equal

- 4 lines of symmetry - 2 lines of symmetry - 2 lines of Symmetry

- Rotational Symmetry 4 - Rotational Symmetry 2- Rotational Symmetry 2

- Diagonal equal in length - Diagonals equal in length - Diagonals not equal in length

-Diagonals bisect at right-angles - Diagonals bisect each other - Diagonals bisect at right angles

- Remember “drunken square”

12

2-d Shapes

Quadrilaterals

Parallelogram Kite Trapezium

- Opposite sides parallel

- Opposite sides equal - 1 line of symmetry - 1 pair of parallel sides

- Opposite angles equal - No Rotational Symmetry - Might have 1 lines of Symmetry

- No lines of symmetry - Diagonals not equal in length - No Rotational Symmetry

- Rotational Symmetry 2 - Diagonals cut at right angles - Diagonals not equal in length

- Diagonal not equal in length

- Diagonals bisect each other

- Remember “drunken rectangle”

13

Calculating Areas

Area is the amount of space inside a FLAT shape.

Area is usually measured in square millimetres (mm2) Very small !!!

square centimetres (cm2) Everyday Shapes

square metres (m2) Floor area in house

square kilometres (km2) Fields or countries?

With irregular shapes, you can usually ESTIMATE the area by counting squares.

Eg.

Estimated area ≈ 5 cm2

Regular shapes will usually have their own area formulae!!

14

Calculating Areas

Rectangle/Square Triangle

Area = length × breadth Area = ½ × base × height

length

breadth

base

height

15

Rhombus/Parallelogram

The rhombus and the parallelogram have the same area formula (much the same way that the square and rectangle use the same formula!)

Area = base × perpendicular height

Calculating Areas

base

height

16

Trapezium

The area of a trapezium could of course be found by splitting it up into smaller triangles and/or rectangles and finding the area piece by piece. Alternatively, the following formula can be used:

Area = ½ ×(sum of the parallel sides) × perpendicular height

Calculating Areas

height

17

The Circle

Parts of the circle:

Radius

Diameter

- A line drawn from the centre of a circle to its edge

- A line drawn from edge to edge of a circle, through its centre

- A line drawn from edge to edge of a circle, NOT through its centre

- The distance around the outside of a circle

- A “pizza slice” of a circle

- A section of the circumference

Chord

Circumference

Sector

Arc

** Note : Diameter = 2 × Radius **

18

The Circle

There are only 2 formulae that you need to learn for circles!!!!

They both include the use of the number

is just a symbol used for the very long number 3.14159 … …

ππ

Circumference of a Circle

Circumference = π × Diameter

Area of a Circle

Area = π × Radius × Radius

C = πD A = πr2

19

3-d Shapes

3-d Shapes are SOLID. This means that you CAN pick them up!

A 3-d shape is NOT described using sides, the way a 2-d shape is.

Instead we discuss :

Faces - a face is a FLAT surface on a 3-d shape

Vertices - a vertex is a corner on a 3-d shape

Edges - an edge is a line where 2 surfaces meet

20

3-d Shapes

Cube Cuboid Sphere Hemi-sphere

- 6 square faces - 6 rectangular faces - No faces - 1 circular face

- 8 vertices - 8 vertices - No vertices - No vertices

- 12 edges - 12 edges - No edges - 1 edge

21

3-d Shapes

Cylinder Cone Triangular-Based Square-Based

Pyramid Pyramid

- 2 circular faces - 1 circular face - 4 triangular faces - 5 faces

- No vertices - 1 vertex - 4 vertices - 5 vertices

- 2 edges - 1 edge - 6 edges - 8 edge

22

3-d Shapes

Prism

A prism is a 3-d shape with 2 identical, parallel bases on which all other faces are rectangular.

Triangular Prism

Heart Shaped Prism

Hexagonal Prism

23

Calculating Volume and Density

Volume

Volume is the amount of space inside a SOLID shape.

Volume is usually measured in cubic millimetres (mm3) Very small – only medicines?

cubic centimetres (cm3) Everyday objects

cubic metres (m3) Volume of a room?

cubic kilometres(km3) Volume of the ocean?

Finding the volume of some objects can be as simple as counting cubes.

Volume = 10 cm3

Most regular shapes however, will have a volume formula.

24

Volume of a Cuboid

Volume = length × breadth × height

Volume of a Prism

Volume = Area of cross-section × length

Note – this formula can also be applied to a cylinder!!!!

Volume of Cylinder = πr 2h

Calculating Volume and Density

25

Density

The density of an object is defined as being its mass per unit volume.

To calculate the density of an object :

Since mass is measured in kg and volume in cm3, then density is measured in kg/cm3.

The triangle below can help you to use and rearrange (when necessary) this formula.

Calculating Volume and Density

VolumeMass

Density

D V

MCover up the letter you want to help you find the right

formula!!

26

Dimensions

The dimension of a formula is the number of lengths that are multiplied together.

A constant has no dimension. It is just a number.

Length has 1 dimension. Any formula for a length can only have constants and a length.

eg. C = π D , P = 2l + 2w

Area has 2 dimensions. Any area formula can only involve constants and length × length.

eg. A = π r2, A = l × b

Volume has 3 dimensions. A volume formula will only involve constants and length × length × length.

eg. V = l × b × h, V = πr2h

27

Dimensions

Some formulae have more than one part.

When this happens, all the different parts of the formula must have the same dimension, or the formula is incorrect.

Eg. A = 2πr2 + 2πrh

This formula is a perfectly acceptable area formula, since both parts have 2 dimensions.

Eg. V = 2πr3 + 2rh

This formula is completely incorrect as a volume formula, since even though the first part does have 3 dimensions, the second part only has 2, making it an area!

28

Angles

Types of Angle

Acute Angle Right Angle Obtuse Angle

(Between 0° and 90°) (Exactly 90°) (Between 90° and 180°)

Straight Angle Reflex Angle Complete Turn

(Exactly 180°) (Between 180° and 360°) (Exactly 360°)

29

Angles

Angles at Parallel Lines

Vertically Opposite AnglesAlternate AnglesCorresponding Angles

(will be EQUAL) (Will be EQUAL) (Will be EQUAL)

(Remember Z shape) (Remember F shape)

a bc

d ef

30

Angles

Angles inside Polygons

• External angles in ANY shape will add to 360°

• Angles in a triangle add to 180°

a + b + c = 180°

• Angles in a quadrilateral add to 360°

a + b + c + d = 360°

• The sum of the interior angles in ANY shape can be found by using the formula

180 (n – 2)

where n is the number of sides

a

a

b

c

a

b c

d

31

Angles

Angles in Circles

• Angle in a semi-circle is ALWAYS a right-angle

•A tangent and radius ALWAYS meet at right-angles

32

Angles

Angles in Circles

• A line drawn from the mid-point of a chord to the centre of a circle is always at right-angles to the chord.

• Opposite angles in a cyclic quadrilateral add to 180°

So :

a + c = 180

and

b + d = 180

a

b

cd

33

Angles

Angles in Circles

• Angles drawn from the same arc are EQUAL

• The angle at the centre is twice the angle at the circumference

So b = 2 × a

a

b

a

b

34

Angles

Bearings

A bearing is an angle.

It is always measured clockwise, starting from North and is always recorded using 3 digits. This means that a bearing of 20° should be recorded as 020°. Using 3 digits means there is less chance of confusion or mistakes!

Bearing of B from A Bearing of A from B

(start at A, facing N and turn to face B)(start at B, facing N and turn to face A)

A

B

A

B

35

Transformations

There are 4 different transformations :

• Translation - A translation is movement in a straight line.

The object being translated will look exactly the same, but its position will change.

• Reflection - The reflection of an object is its mirror image.

The size and shape will stay the same, but the direction will be reversed.

• Rotation - A rotation turns a shape about a fixed point, called the centre of rotation.

• Enlargement - An enlargement changes the size of an object.

36

Transformations

Translation

A translation is usually written as a column vector : eg.

The top number tells us how far ACROSS to move an object (a negative here tells us to go back).

The bottom number tells us how far to move UP (a negative number here means we move down).

54

Starting shape!

2

6

2

7transformation

transformation

transformation

54

37

Transformations

Reflection

When working with a reflection, you must take careful note of the mirror line.

Starting shape!

A B

C

D

Reflection in the line AB

Reflection in the line CD

E

FStarting shape!

Reflection in the line EF

38

Transformations

Rotation

When you describe a rotation, you must give three things - the angle

- the direction (CW or ACW)

- the centre of rotation

Starting shape!

90° clockwise rotation about (0,0)

180° rotation about (-1,2)

39

Transformations

Enlargement

When you describe an enlargement you must give two things

- the centre of enlargement

- the scale factor

When enlarging an object, you are not simply multiplying the length of the sides by the scale factor. Instead, you should multiply the distance from each individual vertex to the centre of enlargement by the scale factor.

Enlargement, Scale Factor 3, Centre (-4,6)

Enlargement, Scale Factor 2, Centre (0,0)

40

Metric Measure

Length

mm cm m km

× 10

× 100 × 1000

÷ 10

÷ 100 ÷ 1000

Capacity

ml l

× 1000

÷ 1000

Mass

mg g kg

× 1000

× 1000

÷ 1000

÷ 1000

41

Metric Measure

Metric ↔ Imperial

Length : cm inches

cm feet

cm yard

m yard

km miles

÷ 2.5× 2.5÷ 30× 30÷ 90× 90÷ 0.9× 0.9÷ 1.6× 1.6

42

Metric Measure

Metric ↔ Imperial

Capacity : ml pints

l pints

l gallons

÷ 600× 600

÷ 0.6× 0.6

÷ 4.5× 4.5

× 8÷ 8