Are nd volume 13 Wht s in Chpter 13? 13 01 re 13 0 Are of circle 13 03 res of trpeziums, kites nd rhomuses 13 04 surfce re of rectngulr prism 13 05 surfce re of tringulr prism 13 06 surfce re of cylinder 13 07 Volume of prism 13 08 Volume of cylinder In this Chpter you will: clculte the res of tringles, qudrilterls nd composite shpes clculte the res of circles, semicircles, qudrnts, sectors, nnuluses nd composite circulr shpes clculte the surfce res of cues, rectngulr prisms nd tringulr prisms clculte the surfce re of cylinder convert etween metric units for volume nd cpcity clculte the volumes of prisms, composite prisms nd cylinders Shutterstock.comS.Pytel ISBN 978017035107 Chpter 13 Are nd volume 59
13 01 Are Are is the mount of surfce spce occupied y flt shpe. Are is mesured in squre units such s squre centimetres (cm ) or squre metres (m ). The tle elow shows the formuls for finding the res of four shpes. Squre Rectngle s w A = (side length) A = s Tringle A = length width A = lw Prllelogrm l h h A = 1 se height A = se height A = h A = 1 h Exmple 1 Find the re of ech shpe. c 6. cm 8.4 m 7.1 cm 10. 5.8 m Solution A = lw A = 1 h c A = h 15.9 cm rectngle tringle prllelogrm = 10. 8 6. = 1 58. 84. = 15.9 7.1 = 66.96 cm = 4.36 m = 11.89 cm 60 Developmentl Mthemtics Book 3 ISBN 978017035107
13 01 Are Exmple Find the re of this composite shpe. 1 cm 11.5 cm Solution This is comined squre nd tringle. Are = Are of squre + re of tringle 1 = 1 + 1 11.5 = 13 cm exercise 13 01 1 Find the re of rectngle with length 1 nd width 11 cm. Select the correct nswer A, B, C or D. A 99 cm B 49.5 cm C 19 D 396 cm Find the re of tringle with se 15 m nd height 1 m. Select A, B, C or D. A 180 m B 90 m C 360 m D 5 m 3 Find the re of ech shpe, to deciml plces if required. c.1 cm 5 m 4. 5.3 cm 1.7 cm d e f 3. m 4 m m 3 cm 1 cm ISBN 978017035107 Chpter 13 Are nd volume 61
exercise 13 01 g 1 m h 8 m 7 cm 9 cm i j 5 m 8 m m 6 cm 3 cm k l 3 cm 4 m 9 m m n 3.6 cm 6.4 cm 4.6 cm 4 Find the re of ech composite shpe. c cm 8 m 1 m 5 cm 10.3 cm 1.3 m 3 m 3 m 10.4 m 10.4 m 10 m 3 m 3 m d 6. e 1 m f 4. cm 4. cm 6.6 m m 3 m 5. mm 8.6 mm 9.5 mm 6 Developmentl Mthemtics Book 3 ISBN 978017035107
13 0 Are of circle To find the re of circle, multiply π y the circle s rdius squred. A = πr mens A = π rdius r If you re given the dimeter, hlve it to find the rdius first. Exmple 3 Find the re of ech circle correct to 1 deciml plce. 6. m 11.4 cm Solution A = πr r = 6. = π 6. = 10.768... 10.8 m A = πr r = 1 11.4 = 5.7 = π 5.7 = 10.0703... 10.1 cm Exmple 4 Find the re of ech shpe correct to deciml plces. 3.4 m Solution.85 cm 14.6 m Are = 1 re of circle 4 1 = π.85 4 = 6.37939... 6.3 Are = 1 re of circle + 14.6 3.4 1 = π 7.3 + 49.64 r = 1 14.6 = 7.3 = 133.3477... 133.35 m ISBN 978017035107 Chpter 13 Are nd volume 63
exercise 13 0 1 Find the re of circle with rdius 4 cm. Select the correct nswer A, B, C or D. A 5.1 cm B 50.3 cm C 1.6 cm D 100.5 cm Find the re of circle with dimeter 6. m. Select A, B, C or D. A 30. m B 19.5 m C 10.8 m D 60.4 m 3 Find the re of ech circle correct to deciml plces. c r = 9 cm r = 4 m d = 1 mm 4 A circulr lwn hs dimeter of 0 m. Find its re, correct to the nerest squre metre. 5 Find, correct to one deciml plce, the re of ech shpe. 4 m 3 m c d e f 6 m 1 cm m 18 m 6 A duck pond is uilt in the shpe of qudrnt. How mny squre metres of lnd (to deciml plces) will it occupy if the rdius of the pond is 3.6 m? istockphotojnsen 64 Developmentl Mthemtics Book 3 ISBN 978017035107
exercise 13 0 7 Find the re of ech shpe, to deciml pces if required. 14 m 6.5 m c 9. d 4. m.6 cm e f 5 cm 3 cm 60 8.4 m 8 A usiness logo is shown elow. A sign is needed for n dvertisement. How much will it cost to mke this shpe from perspex if the perspex costs $8.50 per m? 4.3 m 5 m 3.9 m 5 m 9 This postge stmp ws issued y Semcirculoni to celerte the king s irthdy. Ignore the perfortions, nd ssume n overll semicirculr shpe in your clcultions. Find its re (correct to 1 deciml plce). Forty stmps re printed on ech sheet s shown. i Find the totl re of 40 stmps. ii Wht is the re of the sheet of pper tht 6 1 they re printed on? 6 1 iii How much pper is wsted on ech sheet (shded)? 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 00 mm 6 1 6 1 6 1 6 1 6 1 40 mm 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 160 mm 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 ISBN 978017035107 Chpter 13 Are nd volume 65
13 03 Ares of trpeziums, kites nd rhomuses Trpezium h A = 1 height (sum of prllel sides) A = 1 h( + ) Kite nd rhomus y x A = 1 digonl 1 digonl y x istockphotoolesiilkei A = 1 xy Exmple 5 Find the re of ech qudrilterl. 3.6 cm c 18.4 m Solution 1 m 8 m 5. cm 3 m 9.6 m A = 1 xy A = 1 xy c A = 1 h( + ) kite rhomus trpezium = 1 1 8 = 1 5. 3.6 = 1 3 (18.4 + 9.6) = 48 m = 9.36 cm = 4 m 66 Developmentl Mthemtics Book 3 ISBN 978017035107
exercise 13 03 1 Find the re of kite with digonls nd 14 cm. Select the correct nswer A, B, C or D. A 11 cm B 4 cm C 56 cm D Find the re of trpezium with prllel sides 4 m nd 6 m nd perpendiculr height 11 m. Select A, B, C or D. A 55 m B 110 m C 34 m D 45 m 3 Find the re of ech of the figures elow, to deciml plces if required. A B A B D C D C AC = 0 cm AC =.4 mm BD = 16 cm BD = 1.8 mm c d x y = 3.06 m x = 1.65 m y y x x = 1. y = 0.06 cm e 16. cm f 8. cm 6.8 m 14.5 m 7.4 cm 6.6 m ISBN 978017035107 Chpter 13 Are nd volume 67
exercise 13 03 4 Find the re of ech composite shpe, to deciml plces if required. All dimensions re in centimetres. 8 1 6 1 8 10 11 14 c.8 d 8.9 4.8 8.4 10.8 9.6.1 5 A grden is in the shpe of trpezium s shown nd Elise wnts to plnt roses in it. Find the re of Elise s grden. If ech rose plnt needs 0.5 m to grow, how mny roses cn Elise plnt? c If ech rose plnt costs $4.60, find the cost of uying roses for the grden. 5.8 m 3. m 4.3 m 68 Developmentl Mthemtics Book 3 ISBN 978017035107
13 04 WORDBANK Surfce re of rectngulr prism surfce re The sum of the res of the fces of solid shpe. net The fces of solid shpe lid out flt. The surfce re of solid is esier to clculte y looking t its net. The numer of fces nd their shpe cn then e seen. A cue nd its net A rectngulr prism nd its net h w l s 6 identicl squre fces h l w 6 rectngulr fces, with opposite fces identicl: top nd ottom, front nd ck, left nd right Surfce re of cue SA = 6 (side length) = 6s Surfce re of rectngulr prism SA = ( length width) + ( length height) + ( width height) = lw + lh + wh Surfce re is n re so it is mesured in squre units. Exmple 6 Find the surfce re of ech solid. 9 m 5 m 5 cm 11 m Solution SA = 6 re of squres SA = ottom + front + sides = 6 5 = 11 5 + 11 9 + 5 9 = 150 cm = 398 m ISBN 978017035107 Chpter 13 Are nd volume 69
exercise 13 04 1 Wht is the surfce re of cue with side length of 4 cm? Select the correct nswer A, B, C or D. A 96 cm B 64 cm C 16 cm D 3 cm Wht is the surfce re of rectngulr prism with length 8 m, width 6 m nd height 9 m? Select A, B, C or D. A 174 m B 43 m C 348 m D 864 m 3 Find the re of ech net. 5 cm 4 cm 7 cm c 3 cm 4 cm 4 cm 4 cm 3 cm 4 cm 3 cm 70 Developmentl Mthemtics Book 3 ISBN 978017035107
exercise 13 04 4 Drw the net of ech solid, then find its surfce re. 5 cm 7 m 15 cm 5 Find the surfce re of ech prism, to deciml plces if required. cm 11 cm 9 cm 13 cm 6 cm c d 4 m 1 m 8 m 7 m 6 m e.3 cm f 8.6 cm 7.4 cm 4.5 m 6 Find the surfce re of cue with side length 3.8 mm. Would rectngulr prism with twice the length, the sme width nd hlf the height hve the sme surfce re s this cue? c Check your nswer y finding the surfce re of the rectngulr prism. ISBN 978017035107 Chpter 13 Are nd volume 71
13 05 Surfce re of tringulr prism A tringulr prism nd its net Exmple 7 Find the surfce re of the tringulr prism with this net. 4 cm E 7 cm A B C 7 cm 5 cm 6 cm 5 cm 4 cm D Solution Fces A nd C re identicl rectngles, B is nother rectngle, D nd E re identicl tringles. Surfce re = ( 5 7) + (6 7) + ( 1 6 4) = 136 cm Exmple 8 Find the surfce re of this tringulr prism. re A + re B + re E 10 cm cm 6 cm Solution 5 fces: front nd ck re identicl tringles, the other 3 fces re different rectngles. Surfce re = Are of tringles + re of se + re of LHS + re of RHS = ( 1 6 8) + (6 ) + (8 ) + (10 ) front + ottom + sides = 96 cm 7 Developmentl Mthemtics Book 3 ISBN 978017035107
exercise 13 05 1 Wht re the shpes of the fces of tringulr prism? Select the correct nswer A, B, C or D. A 3 tringles, rectngles C tringles, rectngles 6 cm cm B 1 tringle, 4 rectngles D tringles, 3 rectngles Ech tringulr fce on tringulr prism hs n re of 8 m nd ech rectngulr fce hs n re of 1 m. Wht is th totl surfce re of the prism? Select A, B, C or D. A 48 m B 56 m C 40 m D 5 m 3 Drw the net of ech tringulr prism nd find the surfce re of ech prism. 10 cm 5 cm 6 cm 4 cm 4 Find the surfce re of ech tringulr prism. 3 cm 3 cm 5 cm 1 cm 4 m 3 m 5 m 8 m c 1 cm d 8 m 10 cm 15 cm 17 m 15 m m e f 7. m 3 cm 4. cm 8.4 cm 6 m 4 m 11.4 m 5 This chocolte r hs the shpe of tringulr prism. How mny cm of pper is needed to wrp round the chocolte r? (Assume tht there is no overlp.) 3 cm 1 cm 5 cm ISBN 978017035107 Chpter 13 Are nd volume 73
13 06 Surfce re of cylinder A cylinder nd its net r h r πr h r The net is mde up of identicl circles nd rectngle. The rectngle is formed y cutting the cylinder long one edge (see dotted line) nd opening it up. The length of this rectngle is πr ecuse it is the circumference of the circle nd the width of the rectngle is h, the height of the cylinder. Are of oth circles = πr Are of rectngle = πr h Surfce re of cylinder SA = π rdius + π rdius height = πr + πrh This formul is for closed cylinder, where there is top nd ottom. An open cylinder hs se ut no top so its SA = πr + πrh, only one circle not two. Exmple 9 Find the surfce re of ech cylinder, correct to 1 deciml plce. 3 cm 14.6 m 6.4 m Open cylinder Closed cylinder Solution Surfce re = πr + πrh Surfce re = πr + πrh = π 3 + π 3 8 = π 7.3 + π 7.3 6.4 = 07.345... 07.3 cm = 460.9658... 461.0 m r = 1 14.6 = 7.3 74 Developmentl Mthemtics Book 3 ISBN 978017035107
exercise 13 06 1 Wht re the shpes of the fces of closed cylinder? Select the correct nswer A, B, C or D. A 1 circle, 1 rectngle B circles, 1 rectngle C circles, rectngles D 1 circle, rectngles Find the surfce re of cylinder with rdius 4 cm nd height 5 cm. Select A, B, C or D. A 175.9 cm B 6. cm C 351.9 cm D 301.6 cm 3 Drw the net of cylinder with rdius 6 cm nd height 4 cm. Clculte, correct to two deciml plces, its surfce re. 4 Find, correct to one deciml plce, the surfce re of ech solid. 4. m 8.4 cm c 3 m 8. m 11.5 cm 11 m d open e 4. f.6 m 1.8 m 9. cm 16 cm 10 cm g open 6.6 m 8.4 m 5 The outside of cylindricl wter tnk of height 1.6 m nd dimeter 8.4 m needs to e pinted. The tnk does not hve lid nd the se cnnot e pinted. Find the surfce re of the curved fce of the tnk, correct to two deciml plces. If one cn of pint cn cover 5 m of the tnk, how mny full cns of pint re required? c If one cn costs $64, wht is the totl cost of the pint required? istockphotojodie777 ISBN 978017035107 Chpter 13 Are nd volume 75
13 07 Volume of prism WORDBANK volume The mount of spce occupied y solid shpe, mesured in cuic units such s cuic centimetres (cm 3 ) nd cuic metres (m 3 ). cpcity The mount of liquid or mteril tht continer cn hold, mesured in millilitres (ml), litres (L), kilolitres (kl) nd meglitres (ML). cross-section cross-section A slice tht is cut through prism, prllel to the ends, tht is the sme shpe ll long the prism. Volume nd cpcity 1 cm 3 contins 1 ml 1 m 3 contins 1 kl or 1000 L 1 ml 1 cm 3 1 000 000 = 1 m 3 = 1 kl Exmple 10 Convert: 3700 cm 3 to ml 15 000 cm 3 to L c.4 m 3 to L Solution 3700 cm 3 = 3700 ml 1 cm 3 = 1 ml 15 000 cm 3 = 15 000 ml = (15 000 1000) L 1 L = 1000 ml = 15 L c.4 m 3 =.4 1000 L = 400 L Volume of prism V = re of se or cross-section height V = Ah A h 76 Developmentl Mthemtics Book 3 ISBN 978017035107
13 07 Volume of prism Exmple 11 Find the volume of ech prism. 11 cm 14 cm 3 cm 8.6 m 9. m 6.4 m m Solution Find A, the re of the se or cross-section first. A = 1 14 11 Are of tringle = 77 cm V = Ah = 77 3 h = 3 = 31 cm 3 A = ( 1 9..) + (9. 6.4) Are of tringle + rectngle Tringle s height = 8.6 6.4 =. m = 69 m V = Ah = 69 h = = 138 m 3 exercise 13 07 1 Convert 6. m 3 to L. Select the correct nswer A, B, C or D. A 60 000 B 6 000 C 600 D 6 00 000 Convert 5 L to cm 3. Select A, B, C or D. A 500 000 B 50 000 C 5000 D 500 3 Convert: 0 cm 3 to ml 450 m 3 to L c 7 500 cm 3 to L d 655 ml to cm 3 e 560 L to m 3 f 5.7 kl to m 3 4 Drw cross-section for ech prism. c ISBN 978017035107 Chpter 13 Are nd volume 77
exercise 13 07 5 Find the volume of ech prism, to 1 deciml plce if required. 4 cm 1 cm 10 cm 6 cm 4 cm 9 cm c 8.5 cm.1 cm 6.9 cm d 4. 1.9 cm 6.7 cm e 16 cm 5 cm 16 cm f 10 cm 15 cm 1 cm g 17 cm 4 cm 1 cm 10 cm h 1 9.5 cm 9.5 cm 1 8.7 cm 6 A vse in the shpe of rectngulr prism is long, 5 cm wide nd 16 cm high. Find its cpcity in: cuic centimetres millilitres c litres. 7 A swimming pool hs the shpe of rectngulr prism tht is 11 m long, 5 m wide nd 1.8 m deep. Find the volume of the swimming pool. How mny litres of wter will it hold? Shutterstock.comShmek 78 Developmentl Mthemtics Book 3 ISBN 978017035107
13 08 Volume of cylinder Volume of CYLINDER V = re of circulr se height V = π rdius height V = πr h r h Exmple 1 Find the volume of ech cylinder correct to one deciml plce. 4.5 cm 10.9 m 10. cm 7. m Solution V = πr h = π 4.5 10. = 648.8959... 648.9 cm 3 V = πr h = π 5.45 7. r = 1 10.9 = 5.45 = 671.8547... 671.9 m 3 exercise 13 08 1 Wht is the volume of cylinder with rdius 4 m nd height 1 m? Select the correct nswer A, B, C or D. A 603. m 3 B 50.3 m 3 C 150.8 m 3 D 106.4 m 3 Wht is the volume of cylinder with dimeter 14 cm nd height 6. cm? Select A, B, C or D. A 3817.7 cm 3 B 954.4 cm 3 C 136.3 cm 3 D 1908. 3 3 Find the volume of ech cylinder correct to 1 deciml plce. c 5 cm 14 mm 6 m 10 cm 9 m rdius =.5 mm ISBN 978017035107 Chpter 13 Are nd volume 79
exercise 13 08 d e f 10 cm 4. 100 cm rdius = 5 cm 7.9 cm 13.7 cm 4 A cn of soft drink hs dimeter of 7 cm nd height of 8.5 cm. Wht is the volume of the cn, correct to the nerest 0.1cm 3? Wht is its cpcity in ml? 5 A $1 coin hs rdius of 1 mm nd thickness of 3 mm. Wht is its volume correct to the nerest cuic millimetre? 6 A wter tnk tht is used to store rinwter hs dimeter of 5 m nd height of 6 m. Wht volume (correct to two deciml plces) of wter cn it hold in cuic metres? Wht is its volume, correct to the nerest litre? 7 A stool is supported y three legs in the shpe of cylinders. Ech leg hs rdius of 3 cm nd height of 50 cm. Wht is the volume of the three legs? (Answer correct to the nerest cm 3 ) 80 Developmentl Mthemtics Book 3 ISBN 978017035107
Lnguge ctivity Mix nd Mtch Mtch ech digrm with its description elow. 1 3 1 m 13 m 5 m 10 m 4 5 3 m 6 8 m 9 m 7 8 9 10 11 1 13 A Cue B Squre C Tringulr prism E Net of cue F Rectngle H Surfce re is 360 m M Rectngulr prism O Tringle R Fce S Net of rectngulr prism T Edge U Net of tringulr prism Y Surfce re is 46 m 3 Mtch ech question numer with n nswer letter to decode the nswer to this riddle: Wht do surfce re nd crowd hve in common? 13-3--5 8-1- 4-7-13-3 8 9-6-11 7-1 1-8-10--9 ISBN 978017035107 Chpter 13 Are nd volume 81
Prctice Test 13 Prt A Generl topics Clcultors re not llowed. 1 Evlute 30 6 4 + 0 Simplify 4 + 7 10 3 Complete: 63 : 7 = 1 : 4 Find y. y 100 138 5 Convert 40 to mixed numerl. 6 6 Solve 3(3x 7 ) = 1 7 Does the point (0, 4) lie on the x-xis or y-xis? 8 If x = 7, evlute 8x 9. 9 Evlute: 6 5 7 6 10 If Rvi nd Cindy shre prize of $000 in the rtio : 3, wht is Rvi s shre? Prt B Are nd volume Clcultors re llowed. 13 01 Are 11 Find the re of ech shpe. 4. cm 6.1 m 8.5 m 7. 1.6 m 1 Find the re of rectngle with length 9 cm nd width 3.5 cm. Select the correct nswer A, B, C or D. A 7 cm B 31.5 cm C 5 cm D 1.5 cm 13-0 Are of circle 13 Find the re of circle with rdius 4. m. Select A, B, C or D. A 55.4 m B 6.4 m C 13. m D 1.7 m 14 Find the re of ech shpe, to deciml plces. 8.6 cm 4.3 cm 8 Developmentl Mthemtics Book 3 ISBN 978017035107
Prctice Test 13 13-03 Ares of trpeziums, kites nd rhomuses 15 Find the re of ech shpe. 7.4 m 5 m 6.5 m 3.6 m.1 m 16 Wht is the re of rhomus with digonls 1.8 nd 9. cm? 13-04 Surfce re of rectngulr prism 17 Find the surfce re of ech prism. 3.5 cm 4.1 m 6.8 m.4 m 13-05 Surfce re of tringulr prism 18 Drw the net of this tringulr prism nd then find its surfce re. 5 cm 4 cm 6 cm 1 cm 13-06 Surfce re of cylinder 19 Find the surfce re of ech cylinder correct to 1 deciml plce. 6 cm 6.5 m 11 cm 4.8 m ISBN 978017035107 Chpter 13 Are nd volume 83
Prctice Test 13 13-07 Volume of prism 0 Find the volume of ech prism. 15 cm 13-08 Volume of cylinder 3.5 cm 8. m 7.4 m 1.6 m 1 Find the cpcity of cylindricl swimming pool if its rdius is 3.8 m nd its height is.4 m. Write your nswer: in m 3, correct to three deciml plces correct to the nerest litre. 84 Developmentl Mthemtics Book 3 ISBN 978017035107