Intermediate Tier Shape and space revision

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2 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

3 Angle calculations Angles in a half turn = a 21 0 Angles in a full turn = F angles are equal 57 0 h Angles in a triangle = j 35 0 i Use the rules to work out all angles b Angles in a quadrilateral = Opposite angles are equal e c d k Angles in an isosceles triangle l Z angles are equal f g 42 0 m 8 0

4 Angles and polygons There are 3 types of angles in regular polygons Angles at = 360 the centre No. of sides Exterior = 360 angles No. of sides Interior = e angles e c c c c c c Calculate the value of c, e and i in regular polygons with 8, 9, 10 and 12 sides Answers: 8 sides = 45 0, 45 0, sides = 40 0, 40 0, sides = 36 0, 36 0, sides = 30 0, 30 0, e e e To calculate the total interior angles of an irregular polygon divide it up into triangles from 1 corner. Then no. of x 180 i Total i = 5 x 180 = 900 0

5 Bearings What are these bearings? A bearing should always have 3 figures. A bearing is an angle measured in a clockwise direction from due North Here are the steps to get your answer N Bristol Bath N 56 0 Notice that there is a difference between the outward journey and the return journey What is the bearing of Bristol from Bath? What is the bearing of Bath from Bristol?

6 Units x 1000 x 100 x 10 Learn these metric conversions km kg kl ml cm cg cl mm mg ml Capacity Length Weight Learn these rough imperial to metric conversions Imperial Metric 5 miles 8 km 1 yard 0.9 m 12 inches 30 cm 1 inch 2.5 cm

7 Perimeter Perimeter = 4 x L of a square 6.5m Perimeter = 2(L + W) of a rectangle 2m Circumference = x D of a circle Be prepared to leave answers to circle questions in terms of 26m especially in the non-calculator exam 15cm 5m 31.4m Perim = D + ( x D) 2 Perim = 15 + ( x 15) 2 Circumference Perim = 15 + = 7.5 x D of a semi-circle 2 The perimeter of a shape is the distance around its outside measured in cm, m, etc. 7.2m 18.4m 1m 4.71m 3m 1m 7.85m Perimeter =? = = 14.56m

8 Area formulae The area of a 2D shape is the amount of space covered by it measured in cm 2, m 2 etc. Area of = L x W square Area of = b x h Area of = (a + b) x h Be parallelogram prepared to leave answers Trapezium 2 6m 49m 2 to circle questions 4m in 5mterms of 40m 4m 7m especially in the 10m non-calculator 2 exam 5m Area of = L x W 2m 16m 2 Area of = b x h rectangle triangle 2 Area = ( x r x r) Area 2of = x r 2 18m 2 2m Area circle 9m 8m 9m = ( x 5 x 5) 2 10cm Area = 12.5 Area of = b x h rhombus 24m 2 6m 8m 42m 7m 6m 50.24m 2 3m 7.5m 2 7m 5m

9 Area strategy m 9m 2m What would you do to get the area of each of these shapes? Do them step by step! 2. 7m 2m 10m 3. 4m 4. 8m 5. 3m 6m 6m 1.5m 6m

10 Volume The volume of a 3D solid shape is the amount of space inside it measured in cm 3, m 3 etc. Volume = L x L x L of cube Volume of = Area at end x L a prism 27m 3 3m 4m 56m 3 A = 14m 2 Volume of = L x W x H cuboid Volume of = ( x r 2 ) x L cylinder 2m 42m 3 7m 3m 7m m 3 10m

11 Nets and surface area Cuboid 2 by 2 by 6 Volume = 2 x 2 x 6 = 24cm 3 To find the surface area of a cuboid it helps to draw the net 2 2 4cm cm 2 12cm 2 12cm 2 12cm 2 4cm 2 Net of the cuboid Total surface area = = 56cm 2 Find the volume and surface area of these cuboids: by 4 by 3 6 by 6 by 1 5 by 5 by 5 V = 5 x 4 x 3 = 60cm 3 V = 6 x 6 x 1 = 60cm 3 V = 5 x 5 x 5 = 125cm 3 SA = 94cm 2 SA = 96cm 2 SA = 150cm 2

12 Spotting P, A & V formulae Which of the following expressions could be for: (a) Perimeter (b) Area (c) Volume 4l 2 h 1 r 3 V P 4 r rh 3 A 4 r 2 h A V r( + 3) 1 d 2 4 r + ½r r + 4l 3lh 2 P A P V 4 rl r(r + l) P 4 r r 2 h 3 rl A V A V A

13 Transformations 1. Reflection y Reflect the triangle using the line: y = x then the line: y = - x then the line: x = 1 x

14 Transformations 2. Rotation y Describe the rotation of A to B and C to D When describing a rotation always state these 3 things: No. of degrees Direction Centre of rotation e.g. a rotation of 90 0 anti-clockwise using a centre of (0, 1) B A C x D

15 Transformations 3. Translation What happens when we translate a shape? The shape remains the same size and shape and the same way up it just. slides. Horizontal translation Use a vector to describe a translation 3-4 Vertical translation Give the vector for the translation from.. 1. A to B 2. A to D 3. B to C 4. D to C A C D B

16 Transformations 4. Enlargement y Enlarge this shape by a scale factor of 2 using centre O x O

17 Constructions Have a look at these constructions and work out what has been done Perpendicular bisector of a line 90 0 Triangle with 3 side lengths Bisector of an angle 60 0

18 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 all that grass he can eat A 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 grass he can eat A B 1.5m

19 Similarity Shapes are congruent if they are exactly the same shape and exactly the same size Shapes are similar if they are exactly the same shape but different sizes How can I spot similar triangles? These two triangles are similar because of the parallel lines Triangle C Triangle A Triangle B All of these internal triangles are similar to the big triangle because of the parallel lines

20 Similarity Triangle 2 These two triangles are similar.calculate length y y = = 8.5m Same multiplier x m 15.12m y 7.2m Triangle 1 x 2.1 Multiplier = = 2.1

21 Pythagoras Theorem How to spot a Be prepared to leave your answer Pythagoras question in surd form (most likely in the non-calculator exam) Right angled F 45cm D Hyp 2 = a 2 + b 2 triangle Calculate the size No angles involved in question How to spot the Hypotenuse Longest side & opposite Calculating the Hypotenuse D 21cm DE of DE to 1 d.p.? 2 = DE 2 = Calculating a DEshorter 2 = 45 side DE = 45 F 6cm E ADE = 9 x 5 Calculate the size DE = 3 5 cm of DE in surd form 3cm B 11m?? 16m Calculate the size of AC to 1 d.p. C E Hyp 2 = a 2 + b 2 DE 2 = DE 2 = DE 2 = 2466 DE = 2466 DE = DE = 49.7cm Hyp 2 = a 2 + b = AC = AC = AC = AC = AC = AC AC = 11.6m

22 Pythagoras Questions Look out for the following Pythagoras questions in disguise: y x Find the distance between 2 co-ords Finding lengths in isosceles triangles x x O Finding lengths inside a circle 1 (angle in a semi -circle = 90 0 ) O Finding lengths inside a circle 2 (radius x 2 = isosc triangle)

23 Trigonometry How to spot a Trigonometry question Right angled triangle An angle involved in question Label sides H, O, A Write SOHCAHTOA Write out correct rule Substitute values in If calculating angle use 2 nd func. key Calculating an angle Calculating a side B D 26cm O F 53cm Calculate the size of to 1 d.p. O 11m A H D? H A 73 0 Calculate the size of BC to 1 d.p. C E SOHCAHTOA Tan = O/A Tan = 26/53 Tan = = 0 SOHCAHTOA Sin = O/H Sin 73 = 11/H H = 11/Sin 73 H = m

24 Circle angle theorems A Rule 1 - Any angle in a semi-circle is 90 0 F c B Which angles are equal to 90 0? E D C

25 Big fish?*! Circle angle theorems Rule 2 - Angles in the same segment are equal Which angles are equal here?

26 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 Three radii c Look out for the angle at the centre being part of a isosceles triangle

27 Circle angle theorems Rule 4 - Opposite angles in a cyclic quadrilateral add up to A D C B A + C = and B + D = 180 0

28 A tangent is a line which rests on the outside of the circle and touches it at one point only Circle angle theorems Rule 5 - The angle between the tangent and the radius is 90 0 c

29 Circle angle theorems Rule 7 - Tangents from an external point are equal (this usually creates a kite with two 90 0 angles in.. or two isosceles triangles) 90 0 c 90 0