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6 Chpter Length, re, surfe re n volume Wht you will lern 6A Length n perimeter 6B Cirumferene of irles n perimeter of setors 6C Are of qurilterls n tringles 6D Are of irles 6E Perimeter n re of omposite shpes Keeping in touh with numery 6F Surfe re of prisms 6G Volume of prisms Drilling for Gol exerise 6H Volume of yliners Mths@home: Rinwter tnks Strns: Mesurement n Geometry Sustrns: AREA AND SURFACE AREA VOLUME In this hpter you will lern to: use formuls to lulte res of irles n qurilterls onvert etween re units lulte the re of omposite shpes lulte the surfe re of tringulr n retngulr prisms use formuls to lulte volumes of prisms n yliners onvert etween volume units. This hpter is mppe in etil to the NSW Syllus for the Austrlin Curriulum in the teher resoures t: www.mrige.eu.u/golnsw9 Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

The Millu Viut The Millu Viut in Frne is the tllest rige struture in the worl. Some of the mesurements for the rige re: length:.46 km onrete volume: 80000 m 3 steel les: 1500 tonnes ro/ek re: 70 000 m. Mny of these mesurements re lulte using the shpes tht mke up the struture of the rige. These inlue irles (ross-setion for the min piers) n trpeziums (ross-setion of the rige ek). Aitionl resoures Aitionl resoures for this hpter n e ownloe from Cmrige GO: www.mrige.eu.u/golnsw9 Drilling for Gol: Builing knowlege n skills Skillsheets: Extr prtise of importnt skills Worksheets: Consolition of the topi Chpter Test: Preprtion for n exmintion Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400 93

94 Chpter 6 Length, re, surfe re n volume Pre-test 1 Count the numer of squres in these shpes to fin the res. 4 m m m m 3 m Fin the perimeter of the shpes in Question 1. 3 Count the numer of ues in these solis to fin the volumes. 4 Nme the soli shown on the right. Wht is the nme of the shpe tht is she on top of the soli? 5 Without using lultor, fin the vlue of:.3 10 0.048 1000 70 100 5 134 10 000 e 0.0005 100 000 f 7 160 1000 6 Estimte the re (i.e. numer of squres) in this ellipse. 7 Convert the following to the units shown in the rkets. Rememer: 1 km = 1000 m, 1 m = 100 m n 1 m = 10 mm. 3 m (mm) 0 m (m) 1.6 km (m) 3 mm (m) e 3167 m (km) f 7 m (m) g 0 000 m (km) h 0.03 km (m) m 8 Fin the re of these si shpes, using the given formuls. 3 m A = 1 m h 10 m 4 m A = l A = s 4 m 1.5 m e 3 m 8 m A = 1 h A = h A = l + s 9 Fin the irumferene (C = π r) n re (A = π r ) of this irle, rouning to two eiml ples. f 5 m 5 m 7 km 10 m 4 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 95 Drilling for Gol 6A1 6A 6A Length n perimeter Mesurement is the rnh of mthemtis tht inlues length, perimeter, irumferene, surfe re, volume n pity. We use mesurement when we esign uilings, wter our grens, ontrol stellites, fill our rs with petrol or prtiipte in shool thletis ys. Let s strt: Not enough informtion? All the ngles t eh vertex in this shpe re 90 n the two given lengths re 10 m n 13 m. Is there enough informtion to fin the perimeter of the shpe? If there is enough informtion, fin the perimeter n isuss your metho. If not, wht informtion nees to e provie? Key ies 13 m To onvert etween metri units of length, multiply or ivie y the pproprite power of 10. 1000 100 10 km kilometre 1000 m metre m entimetre 100 10 Perimeter is the istne roun lose shpe. Sies with the sme mrkings re of equl length. 5 m 3 m P = 5 + + 33 = 13 m m mm millimetre 10 m 11 m P = 11 + 4 = 30 m 4 m Stge 5. 5. 5.1 4 Perimeter The totl istne (length) roun the outsie of fi gure Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

96 Chpter 6 Length, re, surfe re n volume Drilling for Gol 6A3 Exerise 6A 1 Fill in the gps on this flow hrt. Write own the vlue of x in these igrms. 6.1 7 x 1000 km 100 m x 10 Exmple 1 Converting units of length Convert to the units shown in the rkets. 5.41 m (mm) 300 m (km) Solution 5.41 m = 5.41 10 = 54.1 mm 300 m = 300 1000 = 3. km Explntion 3 Convert the following length mesurements to the units given in the rkets. 5 m (mm) 41 m (mm).8 m (m) 0.4 m (m) e 4.6 km (m) f 0.9 km (m) g 51 mm (m) h 36 mm (m) i 40 m (m) j 83.7 m (m) k 7000 m (km) l 170 m (km) 1 m = 10 mm n you re moving to smller unit, so multiply. The eiml point ppers to move one ple to the right. 1 km = 1000 m n you re moving to lrger unit, so ivie. The eiml point ppers to move three ples to the left. x Multiply when hnging to smller unit n ivie when hnging to lrger unit. Unerstning 4 A steel em is 8.5 m long n.5 mm wie. Write own the length n the with of the em, in entimetres. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 97 Flueny Exmple Fining perimeters of simple shpes U N SA C O M R PL R E EC PA T E G D ES Fin the perimeter of eh of the following shpes. 5 m 9 m 1 m 4 m 10 m 18 m Solution Explntion P = 10 + 9 + 5 + 4 = 8 m To fin the perimeter, ll the sie lengths together. P = 1 + 18 = 4 + 36 = 60 m There re two lengths of 1 m n two lengths of 18 m. 5 Fin the perimeter of eh of the following shpes. 7m 5 mm 3 mm 7m 5m 7 mm 1 m 6 m 4 m m 9 m 4m 1 m 0 m e f g 5 mm 3 km 6 mm 10 m 8 mm 6 km 7 m h i 1 m 7 m.5 m 5 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

98 Chpter 6 Length, re, surfe re n volume 6A 6 Fin the unknown sie length in these shpes with the given perimeters..1 m 7 mm x m P = 14 m 7 Write own the vlues of the pronumerls in these shpes. 10 m m m Exmple 3 Fining perimeters of omposite shpes Fin the perimeter of this omposite shpe. Solution m 5 m 3 m 7 m Look t the sies opposite the pronumerls. P = ( 5) + 6 + 3 + + 1 = m P = 6.4 m 3 m x m m 1 m Explntion P = 35 mm 8 m 6 m Missing sies re: 5 m 3 m = m 6 m 5 m = 1 m Alterntively, 6 + 5 = m x mm 3.9 m 3 m 6 m 3.1 m m Use tril n error or n eqution to fin the vlue of x; e.g. 5 + 5 + x = 14 Prolem-solving n Resoning 5 m 3 m 5 m 5 m 5.9 m m m 1 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 99 8 Fin the perimeter of eh of the following omposite shpes. 15 m 15 m e 18 m 6 m 6 m 40 m 5 m 30 m 10.7 m 7.4 m 40 m f 8 m 4 m 5 m.3 km 10 m 7 m 1.8 km 1.57 m 15 m First lel ll the missing sie lengths, then fin the perimeter. 9 A lion ge is me up of five stright fene setions. Three setions re 0 m in length n the other two setions re 15.5 m n 3.5 m. Fin the perimeter of the ge. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

300 Chpter 6 Length, re, surfe re n volume 10 Convert the following mesurements to the units given in the rkets. 8 m (mm) 110 000 mm (m) 0.00001 km (m) 0.0 m (mm) e 8 400 m (km) f 6 743 000 mm (km) 11 Fin the perimeters of these shpes. All ngles re right ngles. 0 m 15 m Enrihment: Piture frming 4.1 m 1 A photo 1 m wie n 0 m long is surroune with piture frme 3 m thik. Fin the outsie perimeter of the frme piture. 13 A squre piture of sie lengths 0 m is insie frme of reth x m. Fin the perimeter of the frme piture when: i x = ii x = 3 iii x = 5 Write rule for the perimeter, P, of the frme piture in terms of x. Use your rule to fin the perimeter when x = 3.7..7 m 0 m In prt, first onvert to m, then to mm. 4.7 m 9.3 m x m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 301 Drilling for Gol 6B1 6B Geoger 6B Cirumferene of irles n perimeter of setors The Egyptins, Bylonins n nient Inins foun numer tht links irle s imeter with its irumferene. Toy we ll this numer pi (π) n know it to e 3.14159, orret to five eiml ples. An ext vlue of pi nnot e written own s eiml, s it hs n infinite numer of eiml ples with no pttern. Let s strt: Disovering pi People re still trying to isover more n more eiml ples of pi. Use the internet to fin the urrent worl reor. Key ies Fetures of irle: Dimeter ( ) is the istne ross the entre. Rius (r) is the istne from the entre to the irumferene. The rius is hlf of the imeter. Cirumferene of irle is: C = π r or C = π Use the π utton on your lultor. Speil irle setors: A hlf irle is lle semiirle. A qurter irle is lle qurnt. Some people elieve tht the nient Egyptins use for pi. 7 Cirumferene Rius Setor Dimeter Dimeter A line pssing through the entre of irle with its en points on the irumferene Rius The istne from the entre of irle to its outsie ege Cirumferene The urve ounry of irle Stge 5. 5. 5.1 4 Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

30 Chpter 6 Length, re, surfe re n volume Exerise 6B 1 Nme fetures shown in the irle t right. Wht is the rius of irle if its imeter is 5.6 m? Wht is the imeter of irle if its rius is 48 mm? 3 Write own the rule for the irumferene of irle using: r, the rius, the imeter 4 Determine the frtion of irle shown in these setors. Write the frtion in simplest form. Exmple 4 Fining the irumferene of irle Fin the irumferene of this irle, orret to two eiml ples. Solution C = π r = π 3 = 6π = 18.85 m (to.p.) Explntion Use the formul C = π r or C = π n sustitute r = 3 (or = 6). 6π is the ext nswer. This is eiml pproximtion. 5 Fin the irumferenes of these irles, orret to two eiml ples. Use lultor for the vlue of π. 8 m e 1.1 m 14 m 5 m f 7. m 16. mm 3 m Write the rule C = π r, then sustitute the rius length. Unerstning Flueny Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 303 Exmple 5 Fining the irumferene using the imeter 1 m U N SA C O M R PL R E EC PA T E G D ES Fin the irumferene of this irle, orret to two eiml ples. Solution Explntion C = π = π 1 = 37.70 m (to.p.) Write the formul. C = π is preferre sine is given. Sustitute = 1 n multiply y π. Use lultor n roun your nswer. Note: 37.6991 rouns to 37.70 for two eiml ples. 6 Fin the irumferenes of these irles, orret to two eiml ples. 10 m Use C = π r or C = π. 5 m 3 mm e f 7.4 m 1.8 mm m g h i 10 m 7 mm 14 m Exmple 6 Working with semiirle Fin the perimeter of this semiirle, orret to two eiml ples. 10 m Solution Explntion 1 π + 10 1 = π 10 + 10 = 5.71 m (to.p.) The perimeter onsists of hlf the irumferene of irle (with imeter 10 m) plus the 10 m imeter ross the top. P = Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

304 7 Fin the perimeters of these setors, orret to two eiml ples. 0 4 4 Deie wht frtion of the irumferene you wnt n on t forget to the stright sies. U N SA C O M R PL R E EC PA T E G D ES 6B Chpter 6 Length, re, surfe re n volume (Qurnt).8 m e (Semiirle) f 3.9 m 15.9 km Prolem-solving n Resoning 8 Fin the istne roun the outsie of irulr pool of rius 4.5 m, orret to two eiml ples. 9 Fin the length of string require to surroun the irulr trunk of tree tht hs imeter of 1.3 m, orret to one eiml ple. 10 Give the perimeters of these shpes, orret to two eiml ples. 0 mm m 11.5 m Two semiirles of the sme size mke full irle. Enrihment: The rolling wheel 11 A wheel of rius 30 m is rolle in stright line. Fin the irumferene of the wheel, orret to two eiml ples. 30 m How fr, orret to two eiml ples, hs the wheel rolle fter ompleting: i rottions? ii 10.5 rottions? Cn you fin how mny rottions woul e require to over t lest 1 km in length? Roun your nswer to the nerest whole numer. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 305 Drilling for Gol 6C1 6C Are of qurilterls n tringles In Yer 8 you lernt how to use the formul for the re of rhomus, whih is lso the formul for the re of kite. In Let s strt we re going to use retngle to explin why the formul for kites n rhomuses works. Let s strt There is simple wy to mke kite from retngulr piee of pper. 1 Mke fol line own the mile, from top to ottom, long the xis of symmetry (AC). Mke nother fol line from one sie to the other ut not ross the mile (DB). 3 The she lines in the igrm will e the igonls of the kite. 4 Join A to B, B to C, C to D n D to A to mke the kite. 5 Cut off the four tringles outsie the kite. 6 Use the four tringles to mke nother kite ientil to ABCD. If the originl piee of pper h sies lle x n y: Wht is the re of the retngulr piee of pper? Wht is the re of the kite ABCD? Key ies Conversion of re units: 1000 100 10 km m 1000 100 m 10 mm 10 = 10 10 = 100 100 = 100 100 = 10 000 1000 = 1000 1000 = 1 000 000 1 m 100 m 1 m 1 m 100 100 m m D D A C A C B B Stge 5. 5. 5.1 4 1 squre metre = 10 000 squre entimetres Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

306 Chpter 6 Length, re, surfe re n volume The re of two-imensionl shpe is mesure of the spe enlose within its ounries. Squre Tringle Retngle s h U N SA C O M R PL R E EC PA T E G D ES A = s A= A = Prllelogrm, rhomus Trpezium Rhomus, squre, kite h y h A = h 1 h( x A = 1 xy A= 1 h + ) Exerise 6C Unerstning 1 Copy n omplete: one squre metre 100 m One squre metre = squre entimetres 100 m one squre 1000 m One squre kilometre = squre metres kilometre 1000 m Count the numer of squres to fin the re of these shpes. Eh squre in eh shpe represents 1 squre unit. For prts n e, note tht eh tringle is hlf of retngle. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 307 e 4 f 17 1 3 Nme the shpe tht hs the given re formul. A = l A = 1 xy (3 shpes) A = 1 h A = 1 h( + ) e A = h f A = s Exmple 7 Converting units of re 6 squre rhomus trpezium 9 retngle tringle kite prllelogrm Convert the following re mesurements to the units given in the rkets. 859 mm (m ).37 m (m ) Solution Explntion 859 mm = 859 10 m m mm = 8.59 m 10.37 m =.37 100 m 100 = 3 700 m m m 4 Convert the following re mesurements to the units given in the rkets. m (mm ) 0.4 m (mm ) 500 mm (m ) 310 mm (m ) e.1 m (m ) f 0. m (m ) g 10 000 m (m ) h 3700 m (m ) i 0.001 km (m ) j 4.3 km (m ) k 3 00 000 m (km ) l 39 400 m (km ) Multiply or ivie y: 10 100 or 1000. Flueny Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

308 Exmple 8 Fining res of retngles, tringles n prllelogrms Fin the re of eh of the following plne figures. 5 km 3 m 3m 1.5 m U N SA C O M R PL R E EC PA T E G D ES 6C Chpter 6 Length, re, surfe re n volume 7m 11 km Solution Explntion Are = l =7 3 = 1 m Use the re formul for retngle. Sustitute l = 7 n = 3. Inlue the orret units. 1 Are = h 1 = 11 5 = 7.5 km Use the re formul for tringle. Are = h = 3 1.5 = 4.5 m Use the re formul for prllelogrm. Multiply the se length y the perpeniulr height. Sustitute = 11 n h = 5, where h is the height perpeniulr (t 90 ) to the se. 5 Fin the re of eh of the following plne figures. 8 km 4m 7 m 6m 5 km 1.5 m 1 m Choose from: A = l e f 3m 4. m A = h A = 1 h 4m 5m 9. m Exmple 9 Fining res of rhomuses n trpeziums Fin the re of eh of the following plne figures. 10 mm 4m 3m 9 mm 6m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 309 Drilling for Gol 6C Solution Explntion 6 Fin the re of eh of the following plne figures. 7 mm e Are = 1 xy 1.7 m 6 mm 4. m = 1 10 9 = 45 mm Are = 1 h( + ) = 1 3(4 + 6) = 15 m m 7 m 10 m 3.1 m Use the re formul for rhomus. x n y re the lengths of the igonls. 1 10 9 = 5 9 = 45 Use the re formul for trpezium. Sustitute = 4, = 6 n h = 3. Inlue the orret units. f 11 m 6 m 4 m 9 m 3 m 4.9 km 6. km Choose from: A = 1 xy A = 1 h( + ) Prolem-solving n Resoning 7 A piee of ln hs n re of hlf squre kilometre (0.5 km ). How mny squre metres (m ) is this? 8 A retngulr prk overs n re of 175 000 m. Give the re of the prk, in km. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

310 Chpter 6 Length, re, surfe re n volume 9 Fin the re of the following. e 0.3 m 6 m 3.5 m 0.6 m 1.6 m mm.5 mm 4 mm 1.9 m 1.63 m 3 m 10 An ol piture frme tht ws one squre now lens to one sie to form rhomus. If the istnes etween pirs of opposite orners re 85 m n 1. m, fin the re insie the frme, in m. f 1.51 m 11 Convert the following mesurements to the units given in the rkets. 1.5 km (m ) 0.000005 m (mm ) 75 000 mm (m ) Enrihment: Winows Choose from: A = s A = h A = 1 h A = 1 h( + ) m 1 Six squre winows of sie length m re ple into m 1 m wie y 8.5 m high wll, s shown. The winows re positione so tht the vertil sping etween the 8.5 m winows n the wll eges re equl. The horizontl spings re lso equl. i Fin the horizontl istne etween the winows. 1 m ii Fin the vertil istne etween the winows. Fin the re of the wll, not inluing the winow spes. If the wll inlue 3 rows of 4 winows (inste of rows of 3), woul it e possile to spe ll the winows so tht the horizontl n vertil spings re the sme? (Note: Horizontl sping oesn t hve to e the sme s vertil sping.) Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 311 6D Are of irles We know tht the irumferene of irle is onnete to the imeter y the speil numer pi (π). The re of irle is lso onnete to pi. Let s strt: How oes irle eome retngle? Consier irle ut into equl setors, s shown, then rerrnge to form retngulrstyle shpe. Compre to the irle, wht is the height of the retngle lose to? Compre to the irle, wht is the length of the retngle lose to? Wht oes this sy out the re of the retngle n, hene, the re of irle? How oul the retngle e improve? Key ies r The re of this irulr stge is the prout of pi n the squre of its rius. The formul for the re of irle is:? A = π r A = πr r? Stge 5. 5. 5.1 4 Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

31 Chpter 6 Length, re, surfe re n volume The rius (r) is hlf the imeter ( ). = 10 m r = 5 m A = p 5 Are = 5p m (ext vlue) = 78.539... = 78.54 m (to.p.) A hlf irle is lle semiirle. A = 1 π r A qurter irle is lle qurnt. A = 1 4 π r Exerise 6D 1 Wht is the rius of these irles? 4.1 m 15 m r 10 m r 7.6 m Whih is the orret formul for the re of irle? A A = π B A = 1 π r C A = π r D A = (π r) 3 Wht frtion of full irle is shown y eh of these setors? Unerstning Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 313 Flueny Exmple 10 Fining res of irles Fin the re of this irle, orret to two eiml ples. U N SA C O M R PL R E EC PA T E G D ES 5.1 m Solution Explntion A = π r = π (5.1) = 81.718 = 81.71 m (to.p.) Write the rule n sustitute r = 5.1. 81.718 rouns to 81.71 euse the thir eiml ple is. 4 Fin the re of eh of these irles, orret to two eiml ples. 9 km 6 m Sustitute the rius into A = π r. m e 4.4 mm f 10 m 11.5 km 5 Fin the re of eh of these irles, orret to two eiml ples. First work out the rius. 8.8 m 0 m 4.7 m.6 m e 8.3 km f 7.9 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

314 Chpter 6 Length, re, surfe re n volume 6D 6 Fin the ext re. Exmple 11 Fining res of qurnts Fin the re of this qurnt, orret to eiml ples. Solution A = 1 4 π r 3 m = 1 4 π (3) = 7.0685 = 7.07 m (to.p.) Explntion A qurnt hs one-qurter of the re of full irle. Sustitute r = 3 n evlute. 7 Fin the res of these setors, orret to two eiml ples. 10 m 4 m 10 m 6 m 6.5 m 8 A pizz of rius 15 m is ivie into qurters. Fin the re of eh qurter, orret to the nerest squre entimetre. 10 m Prolem-solving n Resoning 10 m Skillsheet 6A Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 315 9 A irulr rug touhes the eges of squre room of sie length 4 m. Wht is the rius of the rug? Fin the re of the rug, orret to two eiml ples. Fin the re not overe y the rug, orret to two eiml ples. 10 A pizz shop is onsiering inresing the imeter of its fmily pizz try from 3 m to 34 m. How muh igger in re is the new try? 11 You n rerrnge A = π r A to give r =. Use this new rule to fin the rius of π irle for these res. Roun to one eiml ple where neessry. 10 m 117.8 m 4π km Enrihment: Tennis lights 1 A tennis ourt re is lit y four orner lights. The re lose to eh light is onsiere to e goo (G), n the remining re is stisftory (S). Wht perentge of the re is goo? Roun your nswer to the nerest per ent. G G 30 m S Rug 4 m G G 4 m 1 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

316 Chpter 6 Length, re, surfe re n volume 6E Perimeter n re of omposite shpes Composite shpes re me from simple shpes. You n fin the perimeters n res of omposite shpes y strting with simple shpes for whih we hve known formuls. Let s strt: A frtion of irle plus squre This igrm shows frtion of irle plus squre. How oul you fin the re n perimeter of the shpe? Write own full solution for fining the re n perimeter of the shpe. See if your teher or nother stuent n esily follow your solution. Key ies m Composite shpes re me up of more thn one si shpe. Aition n/or sutrtion n e use to fin res n perimeters of omposite shpes. Use ition. Use sutrtion (for re). m 5 m 9 m Exerise 6E 7 m Stge 5. 5. 5.1 4 Unerstning 1 Nme the two ifferent shpes tht mke up eh of these omposite shpes; e.g. squre n semiirle. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 317 Wht frtion of irle is use in eh of these shpes? 3 This omposite shpe inlues squre n qurnt (i.e. 1 4 irle). Fin the re of the squre. Fin the re of the qurnt, orret to two eiml ples. Fin the totl re, orret to two eiml ples. m Fin the perimeter of the entire shpe, orret to two eiml ples. Exmple 1 Fining perimeters n res of omposite shpes Fin the perimeter n re of this omposite shpe, rouning nswers to two eiml ples. Solution P = 3 stright sies + urve sie P = l + + 1 π r = 17 + 14 + 1 π 7 = 69.99 m (to.p.) Retngles: A = l A = 17 14 Are = 38 m Semiirle: A = 1 π r A = 1 π 7 Are = 76.969... m Totl re = 38 + 76.969... = 314.97 m (to.p.) Explntion 3 stright sies + semiirle r Rell tht C = π r. Sustitute l = 17, = 14 n r = 7. Clulte n roun to two eiml ples. The totl re onsists of retngle plus semiirle. 17 m A = l 17 m 14 m + Roun to two eiml ples. Choose from 1 4, 1 n 3 4. For full irle: A = π r C = π r 14 m 7 m A = 1 π r Flueny Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

318 Chpter 6 Length, re, surfe re n volume 6E 4 Fin the perimeter n the re of eh of these omposite shpes, rouning nswers to two eiml ples where neessry. 5 m 3 m 5 m e 6 m 10 m 4 m 1 m 6 m 13 m 1 m 5 m 5 m f 6 mm 8 m m 13 mm Exmple 13 Fining more perimeters n res This shpe is squre with qurnt (qurter irle) sutrte. Fin its perimeter n re, orret to two eiml ples. Solution P = l + 1 4 p r = 8 + 1 4 p 8 = 8.57 m (to.p.) A = s - 1 4 p r = 8-1 4 p 8 = 13.73 m (to.p.) Explntion 4 m First eie whih two shpes you will e eling with. 0 mm Inlues sies n qurter of the irumferene of irle of rius 8 m. 8 m Inlues squre; sutrt qurter of the re of irle of rius 8 m. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 319 5 Fin the perimeters n res of these shpes, orret to two eiml ples. 16 km 4 m 8 km 10 m 6 Fin the re of the she prt of eh of these omposite shpes. e 1 m 8 m 5 m 9 m 10 m 13 m 3 m 4 m 4 m 5 m f 0 m 11 m 3 m 3 m 5 m 0 m Use sutrtion to fin the re. Use sutrtion in eh se. 6 m 6 m 10 m m 6 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

7 An re of lwn is me up of retngle mesuring 10 m y 15 m n semiirle of rius 5 m. Fin the totl re of lwn, orret to two eiml ples. 8 Twenty irulr piees of pstry, eh of imeter 4 m, re ut from retngulr lyer of pstry 0 m long n 16 m wie. Wht is the re, orret to two eiml ples, of pstry remining fter the 0 piees re remove? 9 Fin the re of the she region of eh of the following shpes y sutrting the re of the ler shpe from the totl re. Roun to two eiml ples where neessry. 3 m e 18 m 3 m 6 m 1.7 m 0.6 m 10 m f 9 m Prolem-solving n Resoning 5 m 1.3 m.6 m 3.3 m 15 m 10 m 9.01 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 31 10 Fin the re of eh of the following omposite shpes. 4 m 3 m 4.8 m m 1 m 1.5 m.5 m 1 m Enrihment: Cirulr hllenges 3.9 m 3.5 m 6. m 11 Fin the perimeter n the re of eh of the following omposite shpes, orret to two eiml ples where neessry. 6 m 4 mm m e 1 irle 4 1.8 m f 3 m m 5 m m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

3 Chpter 6 Length, re, surfe re n volume Non-lultor Clultor Keeping in touh with numery 1 Wht is hlfwy etween 1 4 n 1? Wht is hlfwy etween 1 3 n 1? 3 Wht is (-1) 3? 4 Wht is the ue root of 178? 5 How mny kilogrms re there in 3.5 tonnes? 7 Fin: 50% of $400 5% of $400 10% of $400 5% of $400 9 If the short sies of right-ngle tringle re 3 m n 4 m, how long is the hypotenuse? 11 Ameli erns $60 per week. How muh will she ern in yer? 13 Roun these numers, s inite: 1945, to the nerest ten 1995, to the nerest ten 1995, to the nerest hunre 1995, to the nerest thousn 15 Write own five numers tht hve men of 5 n rnge of 5. 17 Consier the lok of ln in the igrm. Fin the perimeter. Fin the re. 9 m 7 m 10 m 5 m 19 Copy n omplete: If ll the sie lengths in Question 17 re oule, the perimeter will e multiplie y. If ll the sie lengths in Question 17 re oule, the re will e multiplie y. 5 m 6 How mny minutes re there in 1 y? 8 Clulte, to the nerest 5: 51% of $417 7% of $417 13% of $417 5.3% of $417 10 Wht is the ext length of the igonl of squre with sies 5 m? 1 Ysmin is on slry of $56 000 per nnum. How muh shoul she e pi eh week? 14 Kren is lying pvers in her retngulr ourtyr, whih is 5.5 m y 4.5 m. She nees 10.9 pvers per squre metre. There re 160 pvers on pllet. How mny pllets will she nee? 16 Steve plye nine holes of golf. His sores were: 6, 6, 5, 5, 7, 10, 6, 7 n 3. Wht ws his totl sore? Wht ws his men sore? Wht ws the rnge? 18 The owners of the lok in Question 17 nee to uil new post-n-wire perimeter fene. They nee to uy post for every orner. Along the sies they nee the posts to e no more thn 3 metres prt. The posts ost $5 eh n the wire osts $7.50 per metre. How muh will this ost? 0 Fin the ost of lying new turf (i.e. grss) on the lok of ln in Question 17 if the turf osts $9.5 per squre metre. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 33 Drilling for Gol 6F1 6F Surfe re of prisms Three-imensionl ojets or solis hve fes tht form the surfe re. Nets re very helpful for etermining the numer n shpes of the surfes of three-imensionl ojet. For this setion we will el with right prisms. A right prism hs uniform rosssetion with two ientil ens n the remining sies re retngles. Right tringulr prism Tringulr ross-setion Let s strt: Builing prisms Use the Drilling for Gol oument to: Print n ut out some nets of prisms. Fol them into 3D solis. Clulte the surfe re. Key ies The Fltiron Builing in New York City is the shpe of tringulr prism. The surfe re of soli is the sum of the res of ll the surfes. A net is two-imensionl illustrtion of ll the surfes of soli. Cue s s s s s A net of ue s s s s s s Surfe re The numer of squre units neee to over the outsie of soli Net A igrm showing how the plne fes of soli re joine to eh other Stge 5. 5. 5.1 4 The formul for the surfe re of ue is: A = 6 s Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

34 Chpter 6 Length, re, surfe re n volume A right prism is soli with uniform ross-setion n with remining sies s retngles. Prisms re nme y the shpe of their ross-setion. The nets for retngulr prism n tringulr prism re shown here. Soli Net Surfe re Retngulr prism l Tringulr prism Exerise 6F 1 How mny fes o the following solis hve? retngulr prism ue tringulr prism Drw net for these prisms n nme eh soli. h l D h A B C A = (l + h + lh ) A = re A + re B + re C + re D Right prism A soli with uniform rosssetion, n remining sies re retngles Unerstning Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 35 3 Copy n omplete the working to fin the surfe re of these solis. 5 m 7 m Exmple 14 Fining the surfe re of retngulr prism Fin the surfe re of this retngulr prism. Solution A = (3 5 + 5 + 3) = 31 = 6 m 4 Fin the surfe re of eh of the following retngulr prisms. Drw net of eh soli to help you. 8 m 4 m 8 m 3 m 7 m m m 3 m A = ( + + ) A = + + + 1 3 m 4 m m Explntion Top Sie Bse 5 m 10 m 3 m 3 m m A up ll the pirs of retngles using A = l. 7 m m e f 5.9 m 3. m.4 m 5 m 3 m m 3 m 4 m 5 m 5.1 m 3. m Flueny Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

36 Chpter 6 Length, re, surfe re n volume 6F Exmple 15 Fining the surfe re of right tringulr prism Skillsheet 1E Fin the surfe re of this tringulr prism. Solution A = 1 5 + 1 4 + 1 3 + ( 1 4 3) = 60 + 48 + 36 + 1 = 156 m 5 Fin the surfe re of eh of the following tringulr prisms. 4 m 5 m 1 m 3 m 3.3 m 7 m 6 m 5 m 4 m 5 m.5 m 1 m 13 m 14 m 5 m 10 m 6 Fin the surfe re of ue with sie length 1 metre. Eh one onsists of three retngles n two tringles. 7 A retngulr ox is to e overe in mteril. How muh mteril is require to over the entire exterior of the ox if it hs the imensions 1.3 m, 1.5 m n 1.9 m? 8 Two wooen oxes, oth with imensions 80 m, 1 m n 5 m, re ple on the groun, one on top of the other, s shown. The entire outsie surfe, inluing the unersie of the ottom ox, is then pinte. Fin the re of the pinte surfe, in m. Only inlue the outsie surfes tht you oul pint. 3 m 5 m Explntion The surfe is me up of three retngles n two ientil tringles. Use 3 m n 4 m for the se n height, respetively, of the tringles. 1 m 1 m 6 m 4 m 7 m Prolem-solving n Resoning 80 m 5 m 5 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 37 9 The four wlls n roof of rn re to e pinte. Fin the surfe re of the rn, not inluing the floor. If 1 litre of pint overs 10 m, fin how mny litres re require to omplete the jo. 10 Cues of sie length 1 unit re stke s shown. Complete this tle. 1 ue ues 3 ues et. Numer of ues (n ) 1 3 4 5 6 7 8 9 Surfe re (A ) Cn you fin the rule for the surfe re (A) for n ues stke in this wy? Write own the rule for A in terms of n. Use your rule to fin the surfe re if there re 100 ues. Enrihment: Pyrmis 3.5 m m.4 m 1.9 m 4 m An exmple of rule is A = n + 5, ut this is not the rule for this question. 11 Pyrmis onsist of se n numer of tringulr fes. The surfe re n e lulte y ing ll the fe res, similr to tht of prisms. Rememer tht the re of tringle is given y A = 1 h. Fin the surfe re of eh of these pyrmis. Drw net of eh soli to help you. 5 mm 3 mm 1.7 m 3 mm 1.8 m 1.7 m 8 m 8 m 10 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

38 Chpter 6 Length, re, surfe re n volume Drilling for Gol 6G1 6G Volume of prisms We use ui units to esrie the mount of spe insie three-imensionl ojet, suh s: ui kilometres for the volume of the Erth ui metres for the spe insie room ui entimetres for the volume of spe oupie y this ook ui millimetres for the volume of metl in pin. Units for pity (millilitres, litres, kilolitres n meglitres) re use for liquis n gses. Let s strt: Why length reth height? For most people, the first thing tht omes to min when eling with volume is length reth height. This rule only pplies to fining the volume of retngulr prisms. Let s look t retngulr prism split into two lyers. How mny ues sit on one lyer? Wht is the re of the se? Wht o you notie? Wht is the height n how mny m lyers re there? 4 m Why is the volume rule given y 3 m 1 m V = lh in this se? 3 m Key ies Common metri units for volume inlue ui kilometres (km 3 ), ui metres (m 3 ), ui entimetres (m 3 ) n ui millimetres (mm 3 ). 1000 3 = 1 000 000 000 100 3 = 1 000 000 10 3 = 1000 For pity, ommon units inlue: meglitres (ML) 1 ML = 1000 kl kilolitres (kl) 1 kl = 1000 L litres (L) 1 L = 1000 ml millilitres (ml) 4 m Stge 5. 5. 5.1 4 1000 3 100 3 10 3 Volume The 1000 3 = 1000 000 000 km 3 m 3 m 3 mm 3 mount of threeimensionl 3 = 1000 000 100 1000 3 100 3 10 3 10 3 = spe 1000 insie n ojet Cpity The mount of liqui ontiner n hol 1000 1000 1000 ML kl L ml 1000 1000 1000 Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry A ui entimetre hols 1 ml of liqui. 1 m 1 m 39 1 m3 hols 1 ml. 1 m U N SA C O M R PL R E EC PA T E G D ES This ontiner hols 1 litre of liqui. 10 m 1000 m3 hols 1 L. 10 m 10 m A ui metre hols 1000 litres or 1 kilolitre. 1m 1 m3 hols 1000 L or 1 kl. 1m 1m Volume of solis with uniform ross-setion is equl to re of ross-setion (A) height (h). V = Ah Cross-setion The plne figure forme when you slie soli figure prllel to one of its surfes Cross-setion h Cross-setion h The height is the length of the ege tht runs perpeniulr to the ross-setion in ny soli. Volume of retngulr prism: h V = lh Exerise 6G Unerstning 1 Wht is the nme given to the shpe of the she ross-setion of eh of the following solis? e f Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

330 Chpter 6 Length, re, surfe re n volume 6G Drw the ross-setionl shpe for eh of these prisms n stte the given height (perpeniulr to the ross-setion). 10 m 6 m 5 mm e 14 m 3 Write the missing numer. The numer of mm in 1 m is. The numer of mm in 1 m is. The numer of mm 3 in 1 m 3 is. There re m 3 in 1 m 3. e There re m 3 in 1 km 3. f There re ml in 1 L. g There re L in 1 kl. h There re m 3 in 1 ml. Exmple 16 Converting units of volume f Perpeniulr mens t right ngle (90 ). m 10 m Choose from: 1000 000 100 10 1000 1 1000 000 000 Convert the following volume mesurements to the units given in the rkets..5 m 3 (m 3 ) 458 mm 3 (m 3 ) Solution Explntion.5 m 3 =.5 100 3 m 3 = 500 000 m 3 100 3 = 1000 000 m 3 m 3 458 mm 3 = 458 10 3 m 3 m 3 mm 3 = 0.458 m 3 10 3 = 1000 458..500000 4 Convert the following volume mesurements to the units given in rkets. 3 m 3 (mm 3 ) 0.3 m 3 (mm 3 ) 000 mm 3 (m 3 ) 0.001 m 3 (m 3 ) e 8.7 m 3 (m 3 ) f 5900 m 3 (m 3 ) g 0.00001 km 3 (m 3 ) h 1 700 m 3 (km 3 ) i 430 000 m 3 (m 3 ) 1 km 3 = 1000 3 m 3 1 m 3 = 100 3 m 3 1 m 3 = 10 3 mm 3 Flueny Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 331 Drilling for Gol 6G t the en of this setion 5 Convert these units of pity to the units given in rkets. 3 L (ml) 0. kl (L) 3500 ml (L) 0.01 L (ml) e 37 000 L (kl) f 4 900 kl (ML) g m 3 (ml) h L (m 3 ) i 1 m 3 (L) Exmple 17 Fining the volume of retngulr prism Fin the volume of this retngulr prism. Solution Volume = lh = 1 1 3 = 3 m 3 Explntion 3 m 1 m 1 m 1 ML = 1000 kl 1 kl = 1000 L 1 L = 1000 ml The soli is retngulr prism. Length = 1 m, reth = 1 m n height = 3 m. 6 Fin the volume of these three-imensionl retngulr prisms. m m m 6 m m 7 m 7 Fin the volume of eh of these retngulr prisms. m m 4 m 3.5 m 3.5 m 3 mm 3.5 m 7 mm 1 m 3 m 5 m 1 mm Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

33 6G Chpter 6 Length, re, surfe re n volume 8 Fin the volume of eh of these three-imensionl ojets. The ross-setionl re hs een given. Simply use Are = m 3 m V = Ah, sine the re of the rosssetion is given. 3.5 m 4.6 m U N SA C O M R PL R E EC PA T E G D ES.6 m m Drilling for Gol 6G3 e f 4.8 m Skillsheet 6G 3.93 m 3.1 m.07 m 5 m 5.8 m Exmple 18 Fining the volume of tringulr prism Fin the volume of this tringulr prism. 3 m 6 m 4 m Solution Explntion 1 h 1 = 4 3 = 6 m Are of ross-setion = The ross-setion is tringle. Note: h = height of tringle. h m V = Ah =6 6 = 36 m3 m h m Multiply the re of the ross-setion y the height of the prism. 9 Fin the volumes of these prisms. 3m 3m 5m 10 m First fin the re of the tringulr ross-setion. 5 m 4 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 333 10 A rik is 10 m wie, 0 m long n 8 m high. How muh spe (volume) woul five of these riks oupy? 11 5 L of wter is poure into retngulr fish tnk tht is 50 m long, 0 m wie n 0 m high. Will it overflow? 1 Fin the volumes of these solis, onverting your nswers to litres. 0 m 40 m 10 m 13 This igrm is sketh of new 5 m swimming pool to e instlle in shool sports omplex. Fin the re of one sie of the pool (she). Fin the volume of the pool, in litres. Use 1 m 3 = 1000 L. 1 m 1.5 m Enrihment: Volume of pyrmi 15 m 6 m 10 m 5 m.5 m 14 Someone tells you tht the volume of pyrmi is hlf of the volume of retngulr prism with the sme se. Do you think this is true? Mke n eute guess s to wht frtion of the prism s volume is the pyrmi s volume. Use the internet to fin the tul nswer to prt. Drw some pyrmis n fin their volume, using the results from prt. Prolem-solving n Resoning There re 1000 m 3 in 1 L. Are of trpezium: A = 1 h( + ) 5 m 10 m.5 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

334 Chpter 6 Length, re, surfe re n volume Drilling for Gol exerise 6G: How mny ues? How mny ues re in eh igrm? Write your nswers on the worksheet or in your exerise ook. 1 3 4 5 6 7 8 Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 335 Drilling for Gol 6H1 6H Volume of yliners A yliner hs uniform ross-setion ( irle), so its volume n e lulte in similr wy to tht of prism. Cylinril ojets re often use to store gses n liquis, so working out the volume of yliner is n importnt mesurement lultion. Let s strt: Five glsses The volume of liqui in this tnker n e estimte using the volume formul for yliner. Fin five ylinril rinking glsses tht re roughly the sme size, ut some re tll n thin n others re short n wie. Lel them rnomly A, B, C, D, E. Everyoy in your lss is require to orer the five letters from smllest volume to lrgest. Key ies The formul for the volume of yliner is: V = π r h where r is the rius of the irulr ens n h is the length or istne etween the irulr ens. Exerise 6H r 1 Fin the re of eh of these irles, orret to two eiml ples. 10 m m h 1.6 m Stge 5. 5. 5.1 4 Unerstning 1.8 km Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

336 6H Chpter 6 Length, re, surfe re n volume Convert the following to the units given in the rkets. Rememer: 1 L = 1000 m3 n 1 m3 = 1000 L. 4.3 m3 (ml) 3.7 L (m3) 000 m3 (L) 3 3 e 38 000 L (m ) f 0.000 m3 (ml) 1 m (L) U N SA C O M R PL R E EC PA T E G D ES 3 Stte the rius (r) n the height (h) of these yliners..6 m 4m 11.1 m 10 m.9 m The height is perpeniulr (t 90º) to the irulr ens. 18 m 1.8 m 3 m e 11.6 m f 10.4 m Skillsheet 6D 15.1 m 1.3 m Flueny Exmple 19 Fining the volume of yliner Fin the volumes of these yliners, orret to two eiml ples. 3 m 1.8 m 10 m 0.3 m Solution V = π r h = π (3) 10 = 90π m3 = 8.74 m3 (to.p.) V = π r h = π (0.9) 0.3 = 0.7634 = 0.76 m3 (to.p.) Explntion Sustitute r = 3 n h = 10 into the formul. The imeter is 1.8 m so r = 0.9. The istne etween the ens is 0.3 m, so h = 0.3. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 4 Fin the volumes of these yliners, orret to two eiml ples. 3 m 1m 337 In prts, e n f, ivie the imeter y to get the rius. 6m U N SA C O M R PL R E EC PA T E G D ES 8 m 7m 8 m.5 m m e f 3.8 m 6m 3m 1. m Prolem-solving n Resoning Exmple 0 Fining the pity of yliner Fin the pity, in litres, of yliner with rius 30 m n height 90 m. Roun your nswer to the nerest litre. Solution Explntion V = π rh Sustitute r = 30 n h = 90. = π (30) 90 = 81 000π Volume = 54 469.004... m3 Every ui m hols 1 ml. Cpity = 54 469.004... ml There re 1000 m3 in 1 L, so ivie y 1000. = 54.469... L = 54 L (to nerest litre) 5 Fin the pity, in litres, of eh of these yliners. Roun to the nerest litre. 15 m 40 m 5 m 100 m First work out the volume in m3. Every 1000 m3 hols 1 litre. 1 m 50 m 10 m 0 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

338 Chpter 6 Length, re, surfe re n volume 6H 6 A ylinril storge rum hs rius of 0.5 m n height of m. Fin its volume, in m 3, orret to three eiml ples. Fin its pity, in L, orret to the nerest litre (1 m 3 = 1000 L). 7 Whih hs greter pity: 10 m y 10 m y 10 m ue or yliner with rius 6 m n height 10 m? 8 A ylinril wter tnk hs rius of m n height of m. Fin its volume, in m 3, roune to three eiml ples. Fin its pity, in L, roune to the nerest litre. 9 How mny litres of gs n tnker rry if its tnk is ylinril with m imeter n is 1 m in length? Roun to the nerest litre. 10 Drw yliner with its irumferene equl to its height. Try to rw it to sle. Enrihment: Cyliner portions 11 Fin the volumes of these ylinril portions, orret to two eiml ples. 0 m m 8 m 5 m 5 m 1 m 15 m 6 m 1 m 1 m 3 = 1000 L First etermine wht frtion of yliner is involve. In prt f, sutrt the missing volume. Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 339 Rinwter tnks In this tivity you will investigte the fesiility of instlling n onneting rinwter tnk to uiling, possily your house or your shool. The intention is tht you will e using the rinwter for vrious purposes, suh s wtering vegetle pth, fighting ushfires or for flushing toilets. You will: Use wesite to ollet t out the frequeny n mount of rin in the re of the uiling. Reserh the ost n pity of tnks tht woul e pproprite for the site. Reserh the osts involve with instlling n mintining the tnk. Estimte how long it woul tke to fill your tnk. Mke preitions out how long your tnk will provie wter for the intene use. Downlo the worksheet from the intertive textook n strt plnning! Filler Art 070 FPO Mths@work 6_Home Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

340 Chpter 6 Length, re, surfe re n volume Puzzles n gmes 1 These towers re me y stking ues. How mny ues re neee in eh se? A lrge ue of sie length 4 m is pinte n then ut into 64 single 1 m ues. How mny 1 m ues re not pinte on ny fe? 3 To the nerest metre, how fr will wheel of imeter 1 m trvel fter 100 revolutions? 4 A got is tethere to the entre of one sie of she with 10 m length of rope. In wht re of grss n the got grze? 5 A irle of rius 10 m hs hole ut out of its entre to form ring. Fin the rius of the hole if the remining re is 50% of the originl re. Roun your nswer to one eiml ple. 6 Here is one net for ue. How mny ifferent nets re possile? Do not ount nets tht n e rotte or reflete to give nother net. 10 m rope 16 m 8 m She 7 A ftory hs flt roof tht hs n re of 100 m. All the wter tht ollets on the roof is fe into rinwter tnk. If there is 1 mm of rinfll, how mny litres of wter go into the tnk? 8 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 341 Units of volume 1000 3 100 3 10 3 km 3 m 3 m 3 3 mm 1000 3 100 3 10 3 1 L = 1000 ml = 1000 m 3 Volume Prisms n yliners V = Ah A = re of ross-setion h = height V = lh V = πr h = 11 6 5 = π() 7 = 330 m 3 = 87.96 m 3 m 5 m 7 m 6 m 11 m Prisms 5 m 3 m 8 m 4 m A = ientil tringles + 3 retngles 1 = 4 3 + 8 5 + 8 4 + 8 3 = 108 m Cirle irumferene C = πr or C = π Length, re, surfe re n volume Surfe re Length n perimeter Cirle 1 m A = πr A = π 6 = 113.10 m = 1 m r = 6 m Are Units of length 1000 100 10 km m m 1000 100 10 Perimeter 10 m 7 m P = 10 + 7 = 34 m Tringle A = 1 h mm Qurilterls Squre A = s Retngle A = l Prllelogrm A = h Trpezium A = 1 h( + ) Rhomus or kite A = 1 xy Units of re 1000 100 10 km m m mm 1000 100 10 h Composite shpes 10 m 8 m A = l 1 πr = 10 8 1 π 4 = 54.87 m P = 10 + 8 + 1 π(4) = 40.57 m Chpter summry Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

34 Chpter 6 Length, re, surfe re n volume Chpter review Chpter tests n worksheets n e foun in the Teher Resoure Pkge. Intertive fill-in-the gps n rg & rop litery tivities n e foun in the Intertive Textook. Multiple-hoie questions An online version of this test is ville in the Intertive Textook. 1 If the re of squre fiel is 5 km, the length of one of its sies is: A 10 km B 5 km C 4 km D 50 km E 5 km If 1 squre metre is 10 000 squre entimetres, then.7 squre metres is: A 70 m B 0.007 km C 7 000 m D 700 mm E 7 m 3 The perimeter of this shpe is: A 1 m B 14 m C 4 m D 0 m 3 m E m 5 m 4 A prllelogrm hs re 10 m n se 5 m. Using A = h, its perpeniulr height is: A 50 m B m C 50 m D m E 0.5 m 5 This shpe is retngle with squre remove. The shpe s re is: A 16 km B 1 km C 0 km D 4 km E 6 km 6 A semiirulr gol re hs imeter 0 m. Its perimeter, orret to the nerest metre, is: A 41 m B 36 m C 83 m D 51 m E 5 m 7 The surfe re of the retngulr prism shown is: A 30 m B 6 m C 31 m D 60 m E 100 m km 4 km 0 m 8 If the perimeter of retngle is 0 m, the lrgest possile re is: A 5 m B 15 m C 5 m D 35 m E 45 m 9 A prism s ross-setionl re is 100 m. The volume is 6500 m 3. The prism s height is: A 0.65 m B 650 000 m C 65 m D 6.5 m E 650 m 10 The volume of yliner with rius 3 m n height 10 m is losest to: A 188 m B 51 m C 141 m 3 D 94 m 3 E 83 m 3 m 3 m 5 km 5 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

343 Short-nswer questions U N SA C O M R PL R E EC PA T E G D ES 1 Convert the following mesurements to the units given in rkets. 1.7 km (m) 48 mm (m) 3.8 m (m) e 5. m (m ) f 0.01 m3 (m3) 73 mm (m ) 3 3 h 3100 ml (L) i 0.043 L (ml) g 53 100 mm (m ) k 4000 m3 (L) l 1 m3 (L) j.83 kl (L) Chpter review Mesurement n Geometry Fin the perimeter of eh of the following shpes. 5m 3m 1.7 m 18 mm 6m 3.6 m 15 mm 4.5 m 3 Fin the re of eh of the following plne figures. 51 mm m 3.7 mm mm 8. mm 5m e m f m 3 m 8 m 3 m 3m 4 Insie retngulr lwn re of length 10.5 m n reth 3.8 m, new gren e is to e onstrute. The gren e is to e the shpe of tringle with se m n height.5 m. Fin the re of the: gren e lwn remining roun the gren e.5 m 3.8 m m 10.5 m Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Chpter 6 Length, re, surfe re n volume 5 Fin the re n irumferene/perimeter of eh of the following shpes, orret to two eiml ples. 1 m 3 m m U N SA C O M R PL R E EC PA T E G D ES Chpter review 344 6 Fin the perimeter n re of eh of the following omposite shpes, orret to two eiml ples. 6m 14.14 m 4m 10 m 7 Fin the surfe re of eh of the following soli ojets. 5m 5 m 3m 1 m 3 m 8.5 m 4m 8 Fin the volume of eh of these soli ojets, rouning to two eiml ples where neessry. Are = 5 m 6m 6 m 3m 3m 10 mm 6 m 8 m Are = 3.14 mm Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400

Mesurement n Geometry 345 Extene-response questions 1 A kinergrten teher ollets some loks of woo for pinting tivity. Eh lok is retngulr prism, s shown. 3 m Fin the volume of eh lok. For eh lok, fin the totl outsie re to e pinte. 5 m If the pint osts $.50 per 100 m, fin the ost of pinting 10 loks. Another woo lok is yliner with rius. m n height 9 m. i Whih lok hs the gretest volume? ii By how muh? (Give your nswer to one eiml ple.) An offie reeives five new esks with enh shpe me 0.8 m up of retngle n qurnt, s shown. The ege of the enh is line with ruer strip t ost 1 m of $.50 per metre. Fin the length of the ruer eging strip for one esk, orret to two eiml ples. Fin the totl ost of the ruer strip for the five esks. Roun your nswer to the nerest ollr. The mnufturer lims tht the esk top re spe is more thn 1.5 m. Is the mnufturer s lim orret? 9 m Chpter review Unorrete 3r smple pges Cmrige University Press Plmer et l, 016 ISBN 978-1-316-61816-5 Ph 03 8671 1400