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1 SHPE, SPE ND MESURES Volume Volume of a cuboid Volume is the amount of space inside a -D shape. he common units for volume are: mm, cm or m. Volume = length x width x height height V = l x w x h V = lwh 5 apacity apacity is the amount of space inside a hollow -D shape. apacity usually refers to the volume of a gas or liquid. You need to know 1000 cm = 1 litre. 50 cm Find the volume of this fish tank, 40 cm giving your answer in litres. 0 cm 6 V = 50 x 40 x 0 = cm V = 60 litres length width his is a net of a cuboid. If one square has an area of 1 cm 2, what is the volume of the cuboid? V = 4 x x 2 = 2 Find the volume of this cuboid. 12 cm Surface area of a cuboid V = lwh = 12 x x 5 = 180 cm. op ip! Substitute numbers into a formula before trying to work anything out. 6 Sample mental test question he volume of a cube is 2 cm. What is the length of an edge of the cube? Since 2 = x x, the length of an edge =. op ip! You should know these cube roots: 1 = 1 8 = 2 2 = 64 = = = 10 here are 6 faces on a cuboid, with opposite faces having the same area. he surface area is given by = 2lw + 2lh + 2wh l w h Sample National est question 6 Find the surface area of the purple cuboid in the panel above. = 2 x 12 x + 2 x 12 x x x 5 = = 222 cm 2. pot heck 1 What is the volume and surface area of this cuboid? 2 cm hese two cuboids have the same volume. Find the value of x. nswer Volume of first cuboid = 6 cm. So volume of second cuboid = 6x = 6 cm. So x = 6 cm. 2 cm he volume of the Sun can hold over a million Earths. x 66 6

2 SHPE, SPE ND MESURES onstructions onstructing triangles When constructing triangles it is very important that you measure lines and angles accurately. onstruct this triangle accurately. op ip! 6 ngle bisector he angle bisector is the line that passes at the same distance from two intersecting lines. onstruct the angle bisector of. cm 8 cm First draw a line 8 cm long. hen use a compass to measure and draw an arc from the left-hand end of the 8 cm line. hen use a compass to measure cm and draw an arc from the right-hand end of the 8 cm line. hen join the ends of the 8 cm line to the point where the arcs cross. Perpendicular bisector Part of the diagram, such as the base line, is often drawn for you. lways use a compass to mark out the distances rather than a ruler. he perpendicular bisector is the line that passes through the midpoint of two other points and is perpendicular (at right angles) to the line that joins them. First set the compass to about. From, draw arcs on and. Where these arcs cross and, draw two further arcs to cross each other. Draw a line from through the point where these arcs cross. his is the angle bisector of. onstructing an angle of 60 Draw an angle of 60 at the point. Set the compass to about. Draw an arc from that crosses the line and draws almost a quarter circle. From the point where the arc crosses the line, draw another arc to cross the first. Join the point where these arcs cross to. he angle at is 60. onstruct the perpendicular bisector of. First set the compass to about two-thirds of the distance from to. Draw arcs from on both sides of the line. Without changing the size of the compass, do the same from. Join the points where the arcs cross. his line is the perpendicular bisector of. op ip! Make sure your arcs are shown. Sample National est question onstruct a triangle that has the following properties: otal length of three sides is 12 cm. Only two of the sides are equal length. ll sides are whole numbers of centimetres. nswer here is only one possible answer. triangle with sides of 2 cm, and. 2 cm bout 4000 years ago, the abylonians tracked the path of the Sun across the sky and realised it took a year (about 60 days) to complete one circuit. his led them to divide the circle into 60. pot heck 1 Draw this triangle accurately cm

3 SHPE, SPE ND MESURES Loci Paths locus (singular of loci) is the path moved by a point according to a rule. Draw the locus of all the points that are a exactly 2 cm from b within of. Loci Most loci problems are set in a real-life context. radio transmitter is to be built so that it is the same distance from two towns: Radville and Seeton. It also has to be within 20 km of a third town, owton. Show the possible location of the transmitter. Same distance from means the perpendicular bisector of. Within 20 km means inside a circle of radius 20 km. he overlap of these two conditions is shown with the red line. 8 a Points that are exactly 2 cm from form a circle of radius 2 cm centred on. b Points that are within of are all points inside a circle of radius centred on. Radville owtown ransmitter could be positioned anywhere on this red line op ip! Make sure your construction arcs are shown and that the required locus is clearly marked. a Draw the locus of all points that are a exactly 2 cm from the line b the same distance from as from. a he points that are exactly 2 cm from form a sausage shape around with two straight lines 2 cm away each side and two semi-circles of radius 2 cm centred on and. b he points that are the same distance from and are the points on the perpendicular bisector of. b b a Sample National Seeton est question Scale: 1 cm represents 10 km he plan shows a garden. Each square is 1 m by 1 m. here are four trees in the garden whose trunks are marked by. John wants to erect an aerial for his short wave radio. he aerial cannot be within 2 metres of any tree trunk nearer than 1 metre to the edge of the garden. Show the places where the aerial could be placed. nswer circle of radius 2 m must be drawn round each tree and all the area within 1 metre of the edge must be excluded. he prohibited areas are shaded. he area that is unshaded is where the aerial could be erected. 8 he locus of the Earth around the Sun is not circular but elliptical. 0 1

4 SHPE, SPE ND MESURES Volume s 5-6 s a What is the volume of this cuboid? Here are four cuboids. cm b What is the surface area? cm 2 2 D 18 cm 12 cm 12 cm 1 cm 2 cuboid has a volume of 6 cm. Its length is 6 cm and its width is. What is the height of the cuboid? cuboid has a volume of 200 cm. Its length and width are. What is the surface area? Remember to include the units in your answer. 4 he volume of a cube is 6. What is the length of each edge of the cube? cm 8 tank has the following measurements. How many litres of water can it hold? Rearrange the cuboids in the order of their volume, with the smallest first. 2 m 80 cm 50 cm cm litres 5 he surface area of this cuboid is Work out the length of the cuboid. 9 hese two cuboids have the same volume. What is the value of x? cm 6 his is a net of a cuboid. What is the volume of the cuboid? 2 m 8 cm 2 cm x m m 1 m cm 66 6

5 SHPE, SPE ND MESURES onstructions s 6- s 6-5 Draw this triangle accurately. 1 onstruct an angle of 60 at the point on the line. 40 cm 8 cm 2 onstruct the perpendicular bisector of the line. onstruct the angle bisector of the angle. 6 Draw this triangle accurately cm 5 4 Draw this triangle accurately. cm 6 cm 68 69

6 SHPE, SPE ND MESURES Loci s -8 4 he diagram shows an island with two airports and. he scale is 1cm represents 10 km. s -8 1 In each of these squares shade the region described. radar station at picks up aircraft within 0 km. a P Q b P Q radar station at picks up aircraft within 40 km. S R S R a ll points that are nearer to P than to Q. b ll points that are nearer to S than to Q. 2 D are squares of side. Match the given loci to the diagrams. Loci a Loci b D D Loci c Loci d a Does the radar station at pick up an aircraft flying directly over? b Show all the points where aircraft are picked up by both radar stations. 5 he diagram shows a garden with a garden shed. Each grid square represents 50 cm. D D i ll points nearer to D than to. ii ll points within of D. iii ll points nearer to the line D than the line. iv ll points within 2 cm of. 4 marks Shed onstruct the locus of the point that is the same distance from the lines and. tree is to be planted. It must not be planted within 1 m of the edge of the garden or the shed. Shade clearly the area in which the tree can be planted. 0 1

7 MHS WORKOOK 5 8 Shape, space and measures answers Pages 66 6 Volume 1 a 1 b 46 cm cm 210 cm 2 ( for units) cm 6 6 m D = 180 cm, = 240 cm, = 288 cm, = l ( for cm or 0.8 m ) 9 Pages onstructions 1 ( for arcs, 1 for accuracy) 2 ( for arcs, 1 for accuracy) ( for arcs, 1 for accuracy) 4 ( for 2 correct sides, for all correct) 5 ( for 1 correct side and 1 angle, for all correct) 6 ( for 1 correct side and 1 angle, for all correct) 1

8 MHS WORKOOK 5 8 Shape, space and measures answers Pages 0 1 Loci 1 a P Q b P Q S R S R 2 i b ii c iii a iv d ( for arcs) 4 a No b Shown half scale 5 Shed 2