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
2 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 b
a
** Notice that the hypotenuse of a right-angled triangle is the longest side and is ALWAYS opposite the right angle. 3 Pythagoras’ Theorem
Example 1 Example 2 finding the hypotenuse finding a shorter side A C
18cm B
B C 21cm A
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.) 4 Trigonometry
Trigonometry is all about finding sides and angles in right-angled triangles.
opp tan adj
opp opposite sin hyp adjacent adj cos hyp
There are a couple of different ways of remembering this: 1) SOH CAH TOA 2) Two Old Angles Skipped Over Heaven Carrying A Harp 5 Trigonometry
Examples : Finding an Angle adj 11.9cm 1) 2) 3) 15.3cm hyp opp 14cm adj hyp opp 21.3cm hyp 19.4cm 9.8cm adj opp
SOH CAH TOA SOH CAH TOA SOH CAH TOA opp opp adj tan sin cos adj hyp hyp 14 9.8 11.9 tan sin cos 19.4 15.3 21.3 14 9.8 11.9 tan1 sin 1 cos 1 19.4 15.3 21.3 35.8 39.8 56.0 6 Trigonometry
Examples : Finding a Side hyp 1) 2) 3) 31.3cm 19° 21.5 hyp x x hyp opp opp x adj opp 41° 63° 15cm adj adj
SOH CAH TOA SOH CAH TOA SOH CAH TOA opp opp adj tan sin cos adj hyp hyp x x x tan 41 sin 63 cos19 15 21.5 31.3 x 15 tan 41 x 21.5sin 63 x 31.3cos19 x 13.0cm x 19.2cm x 29.6c m
7 Trigonometry
Examples : Finding a Side 4) 5) 6) 51° x hyp 14.3cm 4.5cm opp x hyp adj adj opp 63° 73° 19.1cm adj x opp hyp
SOH CAH TOA SOH CAH TOA SOH CAH TOA opp opp adj tan sin cos adj hyp hyp
19.1 14.3 4.5 tan51 sin 63 cos 73 x x x 19.1 14.3 4.5 x x x tan51 sin63 cos73 x 15.5cm x 16.0cm x 15.4cm 8 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 sides Triangle 4 sides Quadrilateral 5 sides Pentagon 6 sides Hexagon 7 sides Heptagon 8 sides Octagon 9 sides Nonagon 10 sides Decagon 12 sides Dodecagon 9 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
10 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” 11 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” 12 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!! 13 Calculating Areas
Rectangle/Square Triangle
breadth height
length base
Area = length × breadth Area = ½ × base × height
14 Calculating Areas
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!)
height
base
Area = base × perpendicular height
15 Calculating Areas
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:
height
Area = ½ ×(sum of the parallel sides) × perpendicular height
16 The Circle
Parts of the circle: Radius - A line drawn from the centre of a circle to its edge Diameter - A line drawn from edge to edge of a circle, through its centre Chord - A line drawn from edge to edge of a circle, NOT through its centre Circumference - The distance around the outside of a circle Sector - A “pizza slice” of a circle Arc - A section of the circumference
** Note : Diameter = 2 × Radius **
17 The Circle
There are only 2 formulae that you need to learn for circles!!!! They both include the use of the number π π is just a symbol used for the very long number 3.14159 … …
Circumference of a Circle Area of a Circle
Circumference = π × Diameter Area = π × Radius × Radius
C = πD A = πr2
18 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
19 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
20 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
21 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 Hexagonal Prism
Heart Shaped Prism
22 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. 23 Calculating Volume and Density
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 24
Calculating Volume and Density
Density The density of an object is defined as being its mass per unit volume. To calculate the density of an object : Mass Density Volume
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.
Cover up the letter you M want to help you find the right formula!! D V
25 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
26 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!
27 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°) 28 Angles
Angles at Parallel Lines
e d c a b f
Vertically Opposite Angles Alternate Angles Corresponding Angles (will be EQUAL) (Will be EQUAL) (Will be EQUAL) (Remember Z shape) (Remember F shape)
29 Angles
Angles inside Polygons • External angles in ANY shape will add to 360° a
• Angles in a triangle add to 180° b a + b + c = 180° a c
• Angles in a quadrilateral add to 360° b c a + b + c + d = 360° a d
• 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 30 Angles
Angles in Circles • Angle in a semi-circle is ALWAYS a right-angle
•A tangent and radius ALWAYS meet at right-angles
31 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° b So : a + c = 180 a and
b + d = 180 d c
32 Angles
Angles in Circles • Angles drawn from the same arc are EQUAL a
b
• The angle at the centre is twice the angle at the circumference
a So b = 2 × a
b
33 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 A
B B
34 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.
35 Transformations
Translation A translation is usually written as a column vector : eg. 4 5
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).
transformation transformation 6 7 2 2
Starting shape!
4 transformation 5 36 Transformations
Reflection When working with a reflection, you must take careful note of the mirror line.
C
Starting shape! Reflection in the line CD
F A B Starting shape!
D
Reflection in the line AB E
Reflection in the line EF 37 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° 180° rotation about (-1,2) clockwise rotation about (0,0)
38 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, Enlargement, Scale Factor 3, Scale Factor 2, Centre (-4,6) Centre (0,0)
39 Metric Measure ÷ 10 ÷ 100 ÷ 1000 Length mm cm m km
× 10 × 100 × 1000
÷ 1000 Capacity ml l
× 1000
÷ 1000 ÷ 1000 Mass mg g kg
× 1000 × 1000
40 Metric Measure
Metric ↔ Imperial
÷ 2.5 Length : cm inches × 2.5
÷ 30 cm feet × 30
÷ 90 cm yard × 90
÷ 0.9 m yard × 0.9
÷ 1.6 km miles × 1.6
41 Metric Measure
Metric ↔ Imperial
÷ 600 Capacity : ml pints × 600
÷ 0.6 l pints × 0.6
÷ 8 × 8
÷ 4.5 l gallons × 4.5
42