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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 Answers: 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 your answer Bristol 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 answers 2 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) 4rl P A r(r + l) Which of the following expressions could be for: A (a) Perimeter (b) Area 1d2 (c) Volume 4r2 4 A 4r3 3 A 3 r + ½r V 4l2h P V 1rh 1r2h 3 A r + 4l 3 1r V P 3 P 4r2h 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 = 35 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 An arrowhead A little fish A mini quadrilateral c Look out for the c angle at the centre being part of a isosceles triangle Three radii Circle angle theorems Rule 4 - Opposite angles in a cyclic quadrilateral add up to 1800 D C A + C = 1800 A and B B + D = 1800 Circle angle theorems Rule 5 - The angle between the tangent and the radius is 900 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