6 Perimeter, re nd volume Syllus topi M. Perimeter, re nd volume This topi will develop your skills to ompetently solve prolems involving perimeter, re, volume nd pity. Outomes Clulte the re of irles nd setors of irle. Clulte the re of omposite figures. Apply the trpezoidl rule to estimte re. Clulte the surfe re of right prisms. Clulte the surfe re of ylinders nd spheres. Clulte the surfe re of pyrmids nd ones. Clulte the volume of right prisms. Clulte the volume of ylinders nd spheres. Clulte the volume of pyrmids nd ones. Clulte the surfe re nd volume of omposite solids. Relte pity to volume. Digitl Resoures for this hpter In the Intertive Textook: Videos Litery worksheet Quik Quiz Widgets Spredsheets Study guide Solutions (enled y teher) In the Online Tehing Suite: Tehing Progrm Tests Tehing Notes Review Quiz Knowledge hek In the Intertive Textook you n tke test of prior knowledge required for this hpter, nd depending on your sore you my e direted to revision from the previous yers work. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
94 Chpter 6 Perimeter, re nd volume 6A 6A Are of irles nd setors The re of irle is lulted using the formul A πr where r is the rdius of the irle. An nnulus is the re etween lrge nd smll irle with the sme entre. The re of n nnulus is lulted using the formul A π ( R r ) where R is the rdius of the lrge irle nd r is the rdius of the smll irle. A setor is prt of irle etween two rdii nd n r. The re of setor is lulted using the formul A θ πr 60 where θ is the ngle formed t the entre of the irle nd r is the rdius of the irle. The re of semiirle A ( π r ) nd the re of qudrnt A ( π r 4 ) re speil ses of the re of setor formul. Are formule Nme Shpe Are Cirle r A πr Annulus R r A π ( R r ) Setor r θ r A θ πr 60 Semiirle r 80 A πr r Qudrnt r r A πr 4 Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6A Are of irles nd setors 95 Exmple : Finding the re of n nnulus 6A Thoms drws two onentri irles whose rdii re 4 m nd 6m. Wht is the re of the nnulus formed, to the nerest squre entimetre? Solution: Drw digrm to represent the informtion in the question. 6 m 4 m The shpe is n nnulus, so use the formul A π ( R r ). Sustitute the vlues for R( R 6) nd r ( r 4). 4 Evlute. 5 Express the nswer orret to the nerest whole numer. 6 Give your nswer with the orret units. 7 Write the nswer in words. A π ( R r ) π (6 4 ) 6.88507 6 m The re of the shpe is 6 m. Exmple : Finding the re of setor 6A Find the re of setor with n ngle t the entre of 55 nd rdius m. Write your nswer orret to one deiml ple. Solution: Drw digrm to represent the informtion in the question. The shpe is setor, so use the formul A θ πr. 60 Sustitute the vlues for θ ( θ 55) nd r ( r ). 4 Evlute. m 55 m A θ πr 60 55 π 60.998677 5 Express the nswer orret to one deiml ple. 6 Give your nswer with the orret units. 7 Write the nswer in words..9 m The re of the shpe is.9 m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
96 Chpter 6 Perimeter, re nd volume 6A Exerise 6A LEVEL Find the re of eh irle. Answer orret to one deiml ple. 4 m 4.5 m. mm d 9. mm e f mm m Wht is the re of the following irles? Answer orret to two deiml ples. Rdius of 5 mm Rdius of m Rdius of 0 m d Dimeter of 8 m e Dimeter of 98 m f Dimeter of.8 mm Exmple Find the re of eh nnulus. Answer orret to one deiml ple. mm 7 m Outer rdius. m Inner rdius 0.6 m 8 mm 6 m d e f.5 mm 5 m m 5 m m 4 mm 4 Wht is the re of eh nnulus? Answer orret to two deiml ples. Outer rdius of 7m nd inner rdius of 4 m Outer rdius of 5.6 m nd inner rdius of. m Outer dimeter of m nd inner dimeter of 8m d Outer dimeter of 44 mm nd inner dimeter of 8 mm Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6A Are of irles nd setors 97 LEVEL 5 Wht is the re of n nnulus with irles of rdii 7 m nd 8.5 m? Answer in squre entimetres orret to one deiml ple. 6 An nnulus onsists of two irles with the sme entre. Find the re of n nnulus tht hs n inner dimeter of 6m nd n outer dimeter of 0 m. Answer orret to the nerest squre entimetre. Exmple 7 Find the re of eh setor. Answer orret to one deiml ple. 0 m m m 9 0 m 6 mm 50 6 mm d 4 m e 0. mm f 5 m 04 5 m 0 4 m 0. mm 8 Find the re of eh setor. Answer orret to two deiml ples. 4 m 4. m 65 5 7 mm d 5 m 4 e. m 80 f 9 m 7 9 Clulte the re of the setor shown opposite. Express your nswer orret to the nerest squre entimetre. 8 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
98 Chpter 6 Perimeter, re nd volume 6A 0 The ross-setion of piee of onduit is n nnulus. It hs n outer dimeter of 60 mm nd n inner dimeter of 50 mm. Find the re of the ross-setion to the nerest squre millimetre. Wht is the inrese in ross-setionl re if the outer dimeter is inresed to 65 mm? Answer orret to the nerest squre millimetre. LEVEL An nnulus with n inner dimeter of 4. metres nd n outer dimeter of 0 metres is divided into two equl prts. Wht is the re of eh prt? Answer orret to three signifint figures. 4. m 0 m The re of irulr pond is 5. squre metres. Wht is the rdius of the pond? Answer orret to one deiml ple. A trffi roundout hs irulr grden in the entre nd two lnes for trffi enirling the grden. The dimeter of the grden is 6 metres nd eh lne is metres wide. Eh lne is to e resurfed. Clulte the re to e resurfed. Answer in squre metres to the nerest whole numer. 4 An nnulus hs n re of 9.70 squre metres. Clulte the inner rdius of the nnulus if the outer rdius is 5 metres. Answer orret to the nerest whole numer. 5 Find the ngle t the entre of these setors to the nerest degree. Are of 04.7 m nd rdius of 0 m Are of 9.6 m nd rdius of 40 m 6 An re to e lndsped is setor with rdius of 9 metres nd n ngle t the entre of 45. The re is to e overed with turf t $.50 per squre metre nd then top-dressed with soil t $.0 per squre metre. Find the totl ost of the lndsping to the nerest dollr. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6B Are of omposite shpes 99 6B Are of omposite shpes A omposite shpe is mde up of two or more plne shpes. The re of omposite shpe is lulted y dding or sutrting the res of eh plne shpe. In some questions it my e neessry to determine the length of missing side. Are of omposite shpes Divide the omposite shpe into two or more plne shpes. Clulte the re of eh plne shpe. Add or sutrt the res of the plne shpes. 4 Write the nswer to the speified level of ury with the orret units. Exmple : Finding the re of omposite shpe 6B A onrete re in the shopping entre onsists of squre nd tringle. The squre hs side length of metres nd the tringle is isoseles. Use Pythgors theorem to find the vlue of x. Clulte the re of the shded region. x Solution: Write Pythgors theorem. Sustitute the length of the sides. Mke x the sujet. 4 Tke the squre root to find x. + x + x x x 7 x 7 m 5 Divide the shded region into squre nd tringle. 6 Use the formul A s to find the re of the squre. 7 Sustitute the vlue for s( s ) into the formul. 8 Evlute. Give your nswer using the orret units. 9 Use the formul A h to find the re of the tringle. 0 Sustitute the vlues for nd h ( 7, h 7). Evlute. Give your nswer using the orret units. Add the res of the two plne shpes to determine the re of the shded region. Express using the orret units. 4 Write the nswer in words. A s (Squre) 44 m A h (Tringle) 7 7 6 m Shded re Are of squre + Are oftringle 44 + 6 80 m Are of shded region is 80 m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
00 Chpter 6 Perimeter, re nd volume 6B Exmple 4: Finding the re of omposite shpe 6B A squre with side length of 6m is ut from tringle with se of 4 m nd height of 6 m. Wht is the shded re? 6 m Solution: The shpe is tringle, so use the formul A h. Sustitute the vlues for nd h ( 4 nd h 6). Evlute nd express using orret units. 4 The shpe is squre, so use the formul A s. 5 Sustitute the vlue for ss ( 6). 6 Evlute nd express using the orret units. 7 Sutrt the re of the squre from the re of the tringle to determine the re of the shded region. 8 Express using the orret units. 9 Write the nswer in words. 6 m 4 m A h (Tringle) 4 6 9 m A s (Squre) 6 6 m Shded re Are of tringle Areofsqure 9 6 56m Shded re is 56 m. Exmple 5: Finding the re of omposite shpe The digrm shows semiirle ut out of semiirle. Wht is the re of the smller semiirle? Wht is the re of the lrger semiirle? Wht is the shded re orret to one deiml ple? Solution: The shpe is semiirle, so use the formul π A r. Sustitute the vlue for r ( r 4). Evlute nd express using orret units. Unless otherwise instruted leve s n ext nswer (8 π ). 4 The shpe is semiirle, so use the formul A πr. 5 Sustitute the vlue for r ( r 8). 6 Evlute nd express using the orret units. 7 Sutrt the res of the two plne shpes to determine the re of the shded region. 8 Write the nswer orret to one deiml ple. 9 Express using the orret units. 0 Write the nswer in words. 4 m 6 m A πr (Semiirle) π 4 8π m 6B A πr (Semiirle) π 8 π m Shded re π 8π 75.9869 75.4 m Shded re is out 75.4 m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6B Are of omposite shpes 0 Exerise 6B LEVEL Exmple Find the re of eh omposite shpe. Answer orret to the nerest whole numer. m 6 m 4 m 5 m 4 m 4 m 4 m 8 m d e f 7 m m 4 m 8 m 8 mm 8 mm 4 m m 8 m A swimming pool is omposite shpe. It is n L-shpe onsisting of two retngles. The dimensions of the L-shpe re shown opposite. Wht is the re of the swimming pool? 6 m 5 m 4 m 8 m A drwing onsists of squre nd tringle. The squre hs side length of 0 m nd the tringle hs sides of length 6m nd 8m. Wht is the re of the drwing? Answer orret to one deiml ple. 0 m 0 m 8 m 6 m Exmple 4 4 A piee of rdord onsists of prllelogrm with two identil squres removed from its shpe. The two squres hve side length of m. Find the shded re. Answer orret to the nerest squre entimetre. m 8 m 5 A irle is removed from squre s shown in the digrm. The squre hs side length of 6m nd the rdius of the irle is m. Wht is the shded re? Answer orret to one deiml ple. m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
0 Chpter 6 Perimeter, re nd volume 6B 6 Find the re of the following irregulr shped loks of lnd. Units re metres. 5 60 60 0 8 50 50 0 50 80 80 45 d D e D 8 f D C 60 50 40 50 45 B C 60 0 48 0 B C 5 6 4 B A A A 7 The digrm on the right shows lok of lnd tht hs een surveyed. All mesurements re in metres. Find the re of the qudrilterl ABCD. Answer orret to one deiml ple. Wht is the length of AB? Answer orret to the nerest metre. C D 5 4 5 B A 8 The digrm on the right shows lok of lnd tht hs een surveyed. All mesurements re in metres. Answer orret to one deiml ple. Find the re of the tringle ABF. Find the re of the tringle ACE. Find the re of the tringle DGE. d Find the re of the trpezium BFGD. e Wht is the totl re of the lok of lnd? C 44 E G 54 67 F 46 5 A B D Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6B Are of omposite shpes 0 LEVEL 9 A omposite plne shpe onsists of semiirle nd n isoseles tringle. Wht is the rdius of the semiirle? Wht is the re of the semiirle? Answer orret to one deiml ple. Clulte the re of the shpe. Answer orret to one deiml ple. mm 0 mm mm 0 A photo n e displyed inside irle using retngulr piture frme whose dimensions re 8 m y 0 m. The irle hs dimeter of 0 m. Wht is the re of the retngulr piture frme? Wht is the re of the irle? Answer orret to two deiml ples. Wht is the re remining fter the irle is removed from the retngulr piture frme? Answer orret to two deiml ples. Exmple 5 A irle is ut from squre with side length of m. Determine the re of the irle. Answer orret to two deiml ples. Wht is the shded re fter the irle hs een removed? Answer orret to two deiml ples. m A sports ground onsists of squre nd two semiirles. Wht is the rdius of the semiirulr ends? Clulte the re of the sports ground in squre metres orret to one deiml ple. 70 m A irle of dimeter 0 m surrounds squre with side length of 5m. Wht is the shded re etween the squre nd the irle? Answer orret to two deiml ples. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
04 Chpter 6 Perimeter, re nd volume 6B LEVEL 4 Luren uts irles with rdius of 4 m from retngulr piee of rdord 8m y 6 m. Wht is the re of the retngulr piee of rdord? How mny irles n e ut from the piee of rdord? Wht is the re of the remining rdord fter the irles hve een ut? Answer orret to two deiml ples. 5 Aron hs deided to lndspe new lwn. The lwn is in the shpe of retngle with semiirle on one end. Explin why the expression for the re of the lwn is π x + xy. Find the re of lwn if x 4 metres nd y 5 metres. Answer orret to the nerest squre metre. y x 6 Oliver wnts to pve retngulr re mesuring.0 m y.5 m in his kyrd. The pvers he wishes to use re 50 m y 50 m. How mny pvers will he need? Answer orret to the nerest whole numer. 7 Snvi is using the omposite shpe on the right s design element. Wht is the re of the semiirle? Answer orret to one deiml ple. Find the height of the tringle. Wht is the re of the omposite shpe? Answer orret to one deiml ple. x mm 4 mm 8 Smnth ought lok of lnd tht is omposite shpe. It onsists of squre with n equilterl tringle on top. The side length of the squre is 40 metres. Wht is the re of the lok of lnd? Answer orret to the nerest squre metre. 9 Aron is using the omposite shpe drwn elow s the sis for grden. The omposite shpe is semiirle with irle of rdius y removed from the entre. Find n expression for the re of the semiirle. Wht is the shded re of the omposite shpe? Find the re of grden if x 6 metres nd y 0.5 metres. Answer orret to the nerest squre metre. x Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6C Trpezoidl rule 05 6C Trpezoidl rule The trpezoidl rule is the re formul for trpezium with different nmes for the vriles. The vrile h is the height or the distne etween the prllel sides, while the vriles d f nd d l re the distnes of the first nd lst prllel sides. Are of trpezium A h ( + ) h ( + ) h Trpezoidl rule A h ( d + d ) f l d f d l h The trpezoidl rule is used to estimte the re of shpe with n irregulr oundry, suh s lke. Exmple 6: Using the trpezoidl rule to estimte the re 6C A lke forms oundry for lok of lnd s shown. Apply the trpezoidl rule to pproximte the re of the lok. Answer to the nerest squre metre 0 m 6 m Lke m Solution: Write the trpezoidl rule A h ( d f + d l ). Write the vlues for h, d f nd d l. Sustitute the vlues for h, d f nd d l nd evlute. 4 Write the nswer using the orret units. A h ( d f + d l ) h 6, df 0, nd d l A 6 (0 +) 756 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
06 Chpter 6 Perimeter, re nd volume 6C Trpezoidl rule A h ( d d ) f + l A Are of shpe h Height or width etween the prllel sides d f Distne long first prllel side d Distne long lst prllel side l d f Irregulr oundry h d l Trpezoidl rule with two pplitions If the trpezoidl rule is pplied more thn one it provides etter estimte for the re of the shpe. Exmple 7: Using the trpezoidl rule to estimte the re 6C Use the trpezoidl rule twie to estimte the re of ABCD to the nerest squre entimetre. B A Setion Setion m 60 m 48 m D 8 m 8 m 6 m C Solution: Divide the re ABCD into two setions t 60. Lel eh setion. Write the trpezoidl rule twie for eh setion: A h d dl ( f + ). Sustitute the vlues for h, d f, nd d l. 4 Evlute. 5 Write the nswer using the orret units. Setion : h 8, d, nd d l 60 Setion : h 8, df 60, nd d l 48 A h ( d f + d ) h l + ( d f + d l ) 8 ( +60) + 8 (60 +48) 70 m f Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6C Trpezoidl rule 07 Exerise 6C LEVEL Exmple 6 Estimte the re y pplying trpezoidl rule one for the following irregulr fields. Answer orret to the nerest squre metre. Billong 6 m 6 m 4 m 8 m River 6 m 4 m 8 m d m m m m Grden 6 m e 4 m f m 0 m m 6 m The following irregulr shpes re the ross-setions for two swimming pools with uniform depth of metres. Clulte the volume of the swimming pools y pplying the trpezoidl rule to estimte the re of the se. 5 m 8 m 4 m 48 m m m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
08 Chpter 6 Perimeter, re nd volume 6C LEVEL The ross-setion of piee of wood is shown opposite. Use the trpezoidl rule to estimte the re of the piee. Answer to the nerest squre entimetre. m 8 m 4 A portion of lnd 6m y 4.8 m is divided into lwn nd ntive grden. Find the re of the lwn y pplying the trpezoidl rule twie. Answer orret to one deiml ple. Wht is the re of the ntive grden? Answer orret to one deiml ple..0 m.0 m 4.8 m Lwn.8 m Ntive grden Exmple 7 5 Connor took three mesurements t 6m intervls ross n irregulr-shped field. The mesurements were 0 m, m nd 8m. Use two pplitions of the trpezoidl rule to lulte the re of the field. Answer orret to the nerest squre metre. LEVEL 6 The digrm opposite shows vertil ross-setion of the oen. It is divided into four vertil strips of equl width y the dshed lines. Estimte the re of the ross-setion y pplying the trpezoidl rule four times. Answer orret to the nerest squre metre. 8 m 0 m m 5 m 7 A uniform ross-setion of sulpture is shown opposite. It is divided into four horizontl strips of equl width y the dshed lines. Apply the trpezoidl rule four times to estimte the re of the ross-setion. Answer orret to the nerest squre metre. 8 m m. m 5 m. m 8 A urved rod hs five offsets to stright oundry line with distnes of 50 m, 60 m, 0 m, 40 m nd 0 m. The offsets re 0 m prt. Use four pplitions of the trpezoidl rule to lulte the re of the field. Answer orret to the nerest squre metre. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6D Surfe re of right prisms 09 6D Surfe re of right prisms To find the surfe re of prism, it is often useful to drw the net of the solid to ensure tht ll the sides hve een dded. A net is plne figure tht shows ll the surfes of solid. When net is folded it forms the solid. The net of prism is identified y eh polygon tht represents surfe. The surfe re (SA) of solid is the sum of the re of eh surfe or polygon of the solid. It is ommon for the re of some of the surfes to e equl. This mkes the lultions esier. Finding the surfe re of right prisms Visulise the surfes of the solid. If neessry drw the net of the solid. Write formul for the surfe re, using the net s guide. Sustitute the vlues into the formul for the surfe re. 4 Use your lultor to find the surfe re. 5 Write the nswer to the speified level of ury with the orret units. Exmple 8: Finding the surfe re of retngulr prism 6D Find the surfe re of this retngulr prism. m 8 m 4 m Solution: Drw net of the retngulr prism. 8 m m 4 m Write down formul for surfe re. Sustitute the vlues into the formul. 4 Evlute. 5 Give nswer to the orret units. 6 Write the nswer in words. SA ( l ) + ( h) + ( l h) SA ( 8 4) + ( 4 ) + ( 8 ) m Surfe re is m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
0 Chpter 6 Perimeter, re nd volume 6D Exmple 9: Finding the surfe re of tringulr prism 6D Find the surfe re of this tringulr prism. 0 mm 0 mm 6 mm Solution: A tringulr prism hs two tringulr fes nd three retngulr fes. The tringulr fe hs se of 8mm nd height of 6mm. Write the formul for the re of tringle. Sustitute the vlues into the re of tringle formul. 4 Evlute. 5 Write n expression for surfe re of the prisms using the re of ll the fes. 6 Evlute nd write nswer to the orret units. 7 Write the nswer in words. 8 mm Tringulr fes A h 8 6 4 mm SA ( 4) + (0 0) + (0 6) + (0 8) 88 mm Surfe re is 88 mm. Exmple 0: Finding the surfe re of trpezoidl prism 6D Find the surfe re of this trpezoidl prism. 6.4 m m 7.7 m Solution: A trpezoidl prism hs two trpezoidl fes nd four retngulr fes. The trpezoidl fe hs perpendiulr height of 6m nd two prllel sides of m nd 8 m. Sustitute the vlues into the formul. 4 Evlute. 5 Write down n expression for surfe re, using the re of ll the fes. 6 Evlute nd write nswer to the orret units. 7 Write the nswer in words. 6 m 6 m 8 m Trpezoidl fes A ( + h ) ( 8 + ) 6 87 m SA ( 87) + ( 6) + (7.7 6) + (8 6) + (6.4 6) 86.6 m Surfe re is 86.6 m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6D Surfe re of right prisms Exerise 6D LEVEL Exmple 8 Find the surfe re of the following retngulr prisms. 0 mm 5 mm 0 mm m m m m 9 m 4 m d 5 mm e f 5 mm 5 mm 4 m m m m 8 m 5 m Find the surfe re of retngulr prisms with the following dimensions. Length 7 m, redth 4 m, height m Length 0 m, redth 9m, height 6m Exmple 9 Find the surfe re of the following tringulr prisms. 4 m 0 mm 0 m 0 mm 6 m 6 m 8 mm 6 mm 6 m d 0 m 6 m 40 m 5 m 6 m 8 m 6 m Are of se is 0.8 sq m e f 0 m 4 m 5 m m 9 m m 5 m m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
Chpter 6 Perimeter, re nd volume 6D LEVEL Exmple 0 4 Find the surfe re of the following right prisms. Are of se is 88 m Are of se is 6 m Are of se is 56 m 5 m 5 m 5 m 4 m 0 m 0 m 5 m 4 m 6 m 4 m 7 m 8 m 0 m 0 m 5 m 7 m d m 5 m 6 m 8 m m e m 4 m 8 m 4 m 7 m 4 m 5 m f 4 m 9 m m 6 m 5 A throom is.5 m long nd.5 m wide nd its wlls re to e tiled to height of.5 m. Find the re of the wlls to e tiled, deduting m for the door nd window. If the tiles ost $50 per m nd the tiler hrges $0 per m to ly them, find the totl ost for tiling the throom wlls. 6 Find the surfe re of the following right prisms. 5 m m m 7.8 m 9 m 5 m. m 0.5 m LEVEL 7 Find the surfe re of right tringulr prism if the tringulr se hs perpendiulr height of m nd se of 8 m. The height of m nd se of 8 m form the two short sides of the tringulr se. The tringulr prism hs length of 0 m. 8 The surfe re of ue is 6.4 m. How mny fes mke up the ue? Find the re of one fe of the ue. 9 Wht is the height of retngulr prism tht hs surfe re of 5 m, given tht the length of the retngulr prism is m nd its redth is 4m? Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6E Surfe re of ylinders nd spheres 6E Surfe re of ylinders nd spheres Cylinders The surfe re (SA) of ylinder is the sum of the re of eh surfe of the solid. To lulte the surfe re of ylinders nd spheres we use the formule elow. Surfe re of ylinder r Open ylinder h SA π rh Curved surfe π rh Closed ylinder r h SA πr + πrh Are of top πr Are of ottom πr Curved surfe πrh Exmple : Finding the surfe re of ylinder A losed ylinder hs dimeter of mm nd height of 50 mm. Find the re of the urved surfe. Find the surfe re of this ylinder. Answer in squre millimetres orret to two deiml ples. mm 6E 50 mm Solution: Write down formul for surfe re of n open ylinder. Sustitute the vlues for r nd h into the formul. Write the nswer orret to two deiml ples. 4 Express using orret units. 5 Write the nswer in words. 6 Write down formul for the surfe re of losed ylinder. 7 Sustitute the vlues for r nd h into the formul. 8 Write the nswer orret to two deiml ples. 9 Express using orret units. 0 Write the nswer in words. SA π rh π 6 50 506.548... 506.55mm Are of the urved surfe is 506.55 mm. SA πr + πrh π 6 + π 6 50 665.04684 665.04 mm Surfe re of ylinder is 665.04 mm. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
4 Chpter 6 Perimeter, re nd volume 6E Surfe re of sphere nd hemisphere The surfe re ( SA ) of sphere is lulted using the formul listed elow. The surfe re of n open hemisphere is hlf tht of the sphere. For losed hemisphere, dd the re of the se, whih is irle. Surfe re of sphere nd hemisphere Sphere r SA 4πr Open hemisphere r SA π r Closed hemisphere r SA π r Are of urved surfe π r Are of se πr Exmple : Finding the surfe re of sphere 6E Wht is the totl surfe re of one dozen tennis lls? Eh ll hs rdius of.5 m. Answer orret to the nerest squre entimetre. Solution: Write down formul for the surfe re of sphere. Sustitute the vlues for r into the formul. Evlute nd express using orret units. 4 Multiply the surfe re of one ll y. 5 Evlute. 6 Write the nswer orret to the nerest whole numer nd express using orret units. 7 Write the nswer in words. SA 4πr 4 π.5 5.9804 m SA lls 5.9804 847.5648 847 m Surfe re of tennis lls is 847 m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6E Surfe re of ylinders nd spheres 5 Exerise 6E LEVEL Exmple Find the surfe re of these open ylinders. Answer to the nerest whole numer. 5 m 6 m mm mm m 8 m Find the surfe re of these losed ylinders. Answer orret to one deiml ple. 9 m 5. m 5 mm 8 mm 9. m 4 m Exmple Find the surfe re of these spheres. Answer orret to two deiml ples. 6 m 5 mm 4 m 4 Find the surfe re of these open hemispheres. Answer orret to two deiml ples.. m 8 m mm 5 Find the surfe re of these losed hemispheres. Answer orret to two deiml ples. m.5 m 7 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6 Chpter 6 Perimeter, re nd volume 6E 6 A n of soup hs 9m dimeter nd is 4 m high. Wht is the rdius of the n? Find the re of lel tht wrps round the n. Answer orret to one deiml ple. Wht is the surfe re of the n inluding the top nd ottom of the n? Answer orret to one deiml ple. LEVEL 7 The outer surfe of spheril ll is mde from plsti. The dimeter of the ll is 0.5 metres nd the ost of plsti is $4 per squre metre. Wht is the outer surfe re of the ll? Answer orret to two deiml ples. Wht is the ost of the plsti for one ll to the nerest ent? Wht is the ost of the plsti for lrger ll whose dimeter is twie the size? 06EQ8 8 Crete the spredsheet elow. LEVEL Cell C6 hs formul tht lultes the surfe re of n open ylinder. Enter this formul. Fill down the ontents of C6 to C using the formul for C6. Cell D6 hs formul tht lultes the surfe re of losed ylinder. Enter this formul. d Fill down the ontents of D6 to D using the formul for D6. e Chnge the rdius of eh ylinder from to 5. f Chnge the height of the tllest ylinder from 8 to 0. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6F Surfe re of pyrmids nd ones 7 6F Surfe re of pyrmids nd ones Surfe re of squre pyrmid The surfe re of pyrmid is found y finding the re of eh fe. For pyrmids, the surfe re is lulted y dding the re of the se nd the re of the tringulr fes. To find the re of the tringulr fe requires the slnt height. The slnt height of right pyrmid is the distne from the pex to the se, long tringulr fe. Apex Slnt height Perpendiulr height Surfe re of squre pyrmid Squre pyrmid hs squre se nd 4 tringulr fes with equl re. s side length of the se l slnt height of tringulr fe s l s SA s + 4 h s + 4 ( sl ) ( ) Exmple : Finding the surfe re of squre pyrmid 6F A squre pyrmid hs se with side length is 6m nd slnt height of 4m. Find the surfe re of this squre pyrmid. Solution: Drw net of the squre pyrmid. 6 m 4 m 6 m 6 m 4 m Write down formul for surfe re. Sustitute the vlues into the formul. 4 Evlute. 5 Express nswer in the orret units. SA s + 4 sl) (6 ) + 4 ( 6 4 84 m ( ) Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
8 Chpter 6 Perimeter, re nd volume 6F Surfe re of one A one hs flt irulr se nd urved surfe. The re of the irulr se is πr nd the re of the urved surfe is πrl where r is the rdius of the irulr se nd l is the slnt height. Apex Slnt height Perpendiulr height Surfe re of one A one hs flt irulr se nd urved surfe. r rdius of the irulr se l slnt height of urved surfe h perpendiulr height of one h l SA πr + πrl r Exmple 4: Finding the surfe re of one A one hs rdius of 6m nd perpendiulr height of 4.4 m. Find the slnt height of the one. Answer orret to one deiml ple. Find the surfe re of the one. Answer orret to the nerest squre entimetre. 6F 4.4 m Solution: 6 m Drw the right tringle nd lel the rdius, perpendiulr height nd slnt height. 4.4 l 6 Write Pythgors theorem. Sustitute the lengths of the sides. 4 Tke the squre root to find l. 5 Write down formul for surfe re. 6 Sustitute the vlues into the formul. 7 Evlute. 8 Express orret to the nerest squre entimetre. l h + r 4.4 + 6 l 4.4 + 6 5.6 SA πr + πrl π 6 + π 6 5.6 407.504... 407 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6F Surfe re of pyrmids nd ones 9 Exerise 6F LEVEL Exmple Find the surfe re of the following squre pyrmids. Answer orret to one deiml ple where neessry. 6. mm 4 m m 6. mm Slnt height is 5 mm 6 m 6 m 8 m d 6. m e f 6 m 6. m 5 m 6. m 5 m 8 m Find the surfe re of the following squre pyrmids. Answer orret to one deiml ple where neessry. 8 m 48 m.5 m 5 m 40 m m A retngulr pyrmid hs se length of.6 m nd redth of 9.4 m The slnt height to the pex from the length is.9 m nd from the redth is.6 m. Wht is the re of the retngulr se? Wht is the re of the tringulr fe tht hs se of.6 m nd height of.9 m? Wht is the re of the tringulr fe tht hs se of 9.4 m nd height of.6 m? d Wht is the surfe re of the retngulr pyrmid?.9 m.6 m.6 m 9.4 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
0 Chpter 6 Perimeter, re nd volume 6F 4 Find the surfe re of the following ones. Answer orret to one deiml ple. 4 m 44 m.5 m 0.95 m 5. m 5 m 5 Find the surfe re of the following ones. Round vlues used in lultions to one deiml ple. Answer orret to the nerest whole numer..8 m. m 8.7 m. m 6.4 m 0.4 m 6 A one hs perpendiulr height of 5.6 m nd slnt height of 7m. Answer these questions orret to one deiml ple. Wht is the rdius of the one? Wht is the re of the irulr se? Wht is the re of the urved surfe? d Wht is the surfe re of the one? 7 m 5.6 m 7 Find the re of the urved surfe for the following ones. Answer orret to the nerest whole numer. Cone with rdius of 9m nd slnt height of 4 m Cone with rdius of 7.4 m nd slnt height of.5 m 8 Find the surfe re of the following ones. Answer orret to the nerest whole numer. Cone with rdius of.7 m nd slnt height of. m Cone with rdius of 6. m nd slnt height of 7.5 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6F Surfe re of pyrmids nd ones LEVEL 9 The solid opposite is lled frustum. It is the portion of squre pyrmid tht remins fter the upper prt hs een ut off y plne prllel to its se. Wht is the surfe re of the frustum? Answer orret to one deiml ple..5 m m 4 m 0 Find the surfe re of retngulr pyrmid with se length of 7m, se redth of 5m nd perpendiulr height of 0 m. Answer orret to signifint figures. The urved surfe re of one is 50 m nd the rdius of the se is 5m. Give nswers orret to one deiml ple. Wht is the slnt height? Wht is the perpendiulr height? Wht is the surfe re of the one? LEVEL A one with rdius of 5m hs surfe re of 00π m. Wht is the perpendiulr height of the one? Answer orret to one deiml ple. A one with se re of 50 m hs totl surfe re of 000 m. Wht is the rdius of the se? Answer orret to two deiml ples. Wht is the slnt height? Answer orret to two deiml ples. Wht is the perpendiulr height? Answer orret to nerest whole numer. 4 Find the surfe re of the following solids. Answer orret to the nerest squre entimetre. 7. m Dimeter 7 m 4 m 5 m 9 m 8.6 m 4.8 m 5 m 8 m m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
Chpter 6 Perimeter, re nd volume 6G 6G Volume of right prisms Volume is the mount of spe oupied y three-dimensionl ojet. It is mesured y ounting the numer of ues tht fit inside the solid. When lulting volume, the nswer will e in ui units. 000 mm m 000 000 m m 000 000 000 m km To lulte the volume of the most ommon solids, we use formul. Some of these formule re listed elow. The volume of prism is found y using its ross-setionl re. Prisms re three-dimensionl ojets tht hve uniform ross-setion long their entire length. Volume of right prisms Cue s s s V Ah ( s ) s s Retngulr prism l h V Ah l h lh Tringulr prism A h H V Ah h H ( ) Exmple 5: Finding the volume of right prism 6G The se re of tringulr prism is 5 m nd the height is 7. m. Wht is the volume of the tringulr prism? Solution: A 5 m 7. m Use the volume formul for right prism V Sustitute the vlues into the formul. Evlute. 4 Write nswer to the orret units. Ah. V Ah 5 7. 08 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6G Volume of right prisms Exmple 6: Finding the volume of retngulr prism A retngulr prism hs length of 8m, redth of m nd height of 4m. Find the volume of this retngulr prism. Answer in ui entimetres. m 6G 4 m 8 m Solution: Use the volume formul for right prism V Ah. Determine the shpe of the se nd the formul to lulte the re of the se A l. Sustitute the vlues into the formul. 4 Evlute. 5 Give the nswer to the orret units. V Ah lh 8 4 64 m Exmple 7: Finding the volume of trpezoidl prism The dimensions of trpezoidl prism re shown opposite. Find the volume of this prism. m 6G 5 m 4 m 5 m 5 m Solution: Find the re of the trpezoidl side (front fe in the digrm.) Use the re formul for trpezium A + ( h. ) Sustitute the vlues into the formul. 4 Evlute. 5 Write nswer to the orret units. 6 Use the volume formul for right prism V 7 Sustitute the vlues into the formul. 8 Evlute. 9 Write nswer to the orret units. Ah. A ( + h ) (+ 5) 4 54 m V Ah 54 5 80 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
4 Chpter 6 Perimeter, re nd volume 6G Exerise 6G LEVEL Exmple 5 Find the volume of the following prisms where A is the re of the se. A 8 m m A 0 m 40 m 6 m A 7 m Wht is the volume of retngulr prism with se re of 5 mm nd height of mm? Find the volume of tringulr prism with height of 5 m nd se re of 50 m. Exmple 6 4 Find the volume of the following solids. Answer to the nerest whole numer. 8 m m 9 m 4 m 8 m 8 m 58 m 7 m 64 m 5 Wht is the volume of retngulr prism with dimensions 4.5 m y 6.5 m y 0.5 m? Answer orret to one deiml ple. 6 Find the volume of the following tringulr prisms, to the nerest whole numers. 0 mm 6 m 0 m 5 m 0 mm 8 mm 6 mm 7 m 9 m m 7 A tringulr prism hs se length of m nd se height of 0 m. Wht is the volume of the prism if the height of the prism is 0 m? Answer orret to one deiml ple. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6G Volume of right prisms 5 Exmple 7 8 Find the volume of the following solids. Answer to the nerest whole numer. 8 m 5 m 7 m 4 m 9 m 6.8 m 0.5 m 8.4 mm. m.5 mm 9 A hollow ontiner is in the shpe of retngulr prism s shown. LEVEL.4 mm 4.6 mm 6 m m 6 m 0 m 0 m Wht would e the volume of the ontiner if it were solid? Wht is the re of the shded se? Wht is volume of the hollow ontiner? 0 A step is shown opposite. Wht is the re of the shded se? Determine the volume of the step. 5 m m 0 m m m LEVEL Find the volume of n equilterl tringulr prism with side lengths m nd depth of 0 m. Answer orret to three deiml ples. A vse with volume of 00 m is pked into the rdord ox shown elow. The spe round the vse is filled with fom to protet the vse from reking. The prel is seled nd posted. 6 m 0 m 8 m Wht is the volume of the fom? Wht is the re of rdord on the surfe of the ox? Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6 Chpter 6 Perimeter, re nd volume 6H 6H Volume of ylinders nd spheres Cylinders A ylinder is right prism whose ross-setion is irle. Its volume is found using V Ah where A πr. The volume of ylinder is the produt of π, the squre of the rdius nd the height. Volume of ylinder r Cylinder h V Ah ( πr ) h πr h Exmple 8: Finding the volume of ylinder A ylinder hs rdius of 8mm nd height of mm. Find the volume of the ylinder. Answer orret to two deiml ples. mm 6H 8 mm Solution: Use the volume formul for ylinder V πr h. Sustitute the r 8 nd h into the formul. Evlute. 4 Write the nswer orret numer of deiml ples nd with orret units. V πr h π 8 4.74 58 4.74 mm Exmple 9: Finding the volume of nnulus prism 6H The dimeter of DVD is m, nd the dimeter of the hole in its entre is.5 m. Find its volume in ui entimetres, orret to two deiml ples, given tht the DVD is 0. m thik. Solution: Use the re formul for n nnulus A π ( R r ). Sustitute the R 6 nd r 0.75 into the formul. Evlute. 4 Use the volume formul for right prism V Ah. 5 Sustitute the A... nd h 0. into the formul. 6 Evlute. 7 Write nswer orret to two deiml ples. A π ( R r ) π (6 0.75 ).08 m V Ah.08 0..596.6 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6H Volume of ylinders nd spheres 7 Volume of sphere The volume of sphere is four-thirds the produt of π nd the ue of the rdius. The volume of hemisphere is hlf the volume of the relted sphere or four-thirds the produt of π nd the ue of the rdius. Volume of sphere Sphere r V 4 π r Hemisphere r V 4 πr πr Exmple 0: Finding the volume of sphere nd hemisphere 6H Find the volume of these solids. Answer orret to one deiml ple. 7 m.5 m Solution: Use the volume formul for sphere V 4 r π. Sustitute r 7 into the formul. Evlute. 4 Write the nswer orret to one deiml ple nd with orret units. 5 Use the volume formul for hemisphere. 6 Find r nd sustitute the vlue into the formul. 7 Evlute. 8 Write the nswer orret to one deiml ple nd with orret units. V 4 πr 4 π 7 46.75504 46.8 m ( V 4 πr πr r.5.75 m V π.75.46. m ) Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
8 Chpter 6 Perimeter, re nd volume 6H Exerise 6H LEVEL Exmple 8 Find the volume of the following ylinders. Answer to the nerest whole numer. m 4 mm.5 m 6 mm d 9 m 7 mm 4 m 0 mm e f 6 m m 0 m 8 m A losed ylindril plsti ontiner is 0 m high nd its irulr end surfes eh hve rdius of 5m. Wht is its volume, orret to two deiml ples? Exmple 0 Find the volume of the following solids. Answer to the nerest whole numer. 5 m 8 mm 4 m d e 5 m f. m 80 mm 4 Find the volume of sphere with dimeter of mm. Answer orret to the nerest whole numer. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6H Volume of ylinders nd spheres 9 LEVEL 5 Find the volume of the following solids. Answer orret to one deiml ple. m 6.8 m m. m 8 m 6 m 6 A wter tnk is in the shpe of losed ylinder with rdius of 0 m nd height of 8m. 0 m 8 m Wht is the re of the top irulr fe of the wter tnk? Leve your nswer in terms of π. Determine the volume of the wter tnk. Leve your nswer in terms of π. 7 A ylindril whet hopper is 8m ross nd 0 m high. Clulte its volume orret to one deiml ple. Grin pours out t 48 000 m per seond. How long would it tke full hopper to empty? Answer orret to the nerest seond. 8 The irumferene of Erth t the equtor is out 40 075 km. Use the formul C π r to find the rdius of Erth orret to the nerest kilometre. Use this rdius to find the volume of Erth. Express your nswer in stndrd nottion using three signifint figures. 9 A ylinder hs its rdius douled. How mny times is its volume inresed? LEVEL 0 A n of tennis lls ontins three lls squeezed in with no room for the lls to move. The ontiner hs height of m. Answer these questions orret to one deiml ple. Wht is the rdius of tennis ll? Clulte the volume of ll. Clulte the volume of the ontiner. d Wht volume of the ontiner is not oupied y the lls? e Express this unused spe s perentge of the totl ontiner. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
0 Chpter 6 Perimeter, re nd volume 6I 6I Volume of pyrmids nd ones A pyrmid fits extly inside prism nd oupies one third of the volume of the prism. Therefore the formul for pyrmid will e V Ah where A is the re of the se nd h is the height of the pyrmid. Likewise, the volume of one is one third the volume of ylinder. Volume formule Nme Solid Volume Squre pyrmid s h V Ah s h s Retngulr pyrmid h V Ah l h l Tringulr pyrmid H A V AH h H Cone h r V Ah πr h Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6I Volume of pyrmids nd ones Exmple : Finding the volume of squre pyrmid 6I Find the volume of this squre pyrmid of height 6m nd se 5m. Give your nswer orret to the nerest ui metre. 6 m 5 m 5 m Solution: Use the volume formul for pyrmid V Ah. Determine the shpe of the se nd the formul to lulte the re of the se: A s. Sustitute the vlues into the formul. 4 Evlute. 5 Give the nswer to the orret units. 6 Write the nswer in words. V Ah s h 5 6 50 m Volume of the pyrmid is 50 m. Exmple : Finding the volume of one 6I Find the volume of this one. Answer orret to three signifint figures. Solution: Use the volume formul for one V Ah. Determine the shpe of the se nd the formul to lulte the re of the se: A πr. Sustitute the vlues for r nd h into the formul. 4 Evlute. 5 Express the nswer in the orret units. 6 Write the nswer in words. V 5 m m Ah πr h π 5 4.59654... 4 m Volume of the one is 4 m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
Chpter 6 Perimeter, re nd volume 6I Exerise 6I LEVEL Exmple Find the volume of the following pyrmids. Answer to the nerest whole numer. 0 m A 49 m 6 m A 7 m 4 m A 9 m d 0 m e 4 m f 8 m 9 m 0 m 6 m 9 m m 6 m Find the volume of the solids with the following dimensions. Give the nswers orret to the nerest whole numer. Retngulr pyrmid with se re 4 m nd height 0 m. Tringulr pyrmid with se re 0 mm nd height 50 mm. A squre pyrmid hs se side length of m nd height of 4 m. Wht is its volume? Answer orret to the nerest ui entimetre. Exmple 4 Find the volume of these ones to the nerest whole numer. 8 m 7 m 4 m 5 m 9 m 4 m 5 Find the volume of one with se rdius 4.5 m nd height 4 m. Answer orret to two deiml ples. 4 m 4.5 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6I Volume of pyrmids nd ones LEVEL 6 The Red Pyrmid is the third tllest pyrmid ever uilt. It hs squre se with side length of 0 m nd height of 04 m. Wht is the re of the squre se? Determine the volume of the Red Pyrmid. Answer orret to two deiml ples. 7 An ie-rem one mesures 4.6 m in dimeter nd is 9m high. Clulte the volume of the one. Answer orret to two deiml ples. 8 A tringulr pyrmid hs se with height of 5 m nd se length of 5m. Wht is the re of the se of the pyrmid? Wht is the volume of the tringulr pyrmid if the height of the pyrmid is m? LEVEL 9 A tringulr pyrmid hs se with sides 6 m, 7 m nd 7 m. The height of the pyrmid is 0 m. Wht is the volume of the tringulr pyrmid? Answer orret to two deiml ples. 6 m 0 m 7 m 0 A frustum is portion of solid. The frustum shown opposite is formed from one. Wht is the volume of the missing top one? Give nswers orret to the nerest ui metre. Wht ws the volume of the entire one? Wht is the volume of the frustum? 0 m 5 m 0 m 0 m Find the volume of the omposite solid shown opposite. This solid onsists of the retngulr prism nd retngulr pyrmid. Answer orret to the nerest ui entimetre. m m m 4 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
4 Chpter 6 Perimeter, re nd volume 6J 6J Surfe re nd volume of omposite solids Surfe re of omposite solids The surfe re of omposite solid is lulted in the sme wy s for right prisms nd pyrmids. Visulise the surfes of the solid, lulte the re of eh fe nd then dd these res together. Surfe re of omposite solid Visulise the surfes of the solid. Write formul nd lulte the surfe re of eh fe. Add the re of eh surfe. 4 Chek tht ll surfes hve een ounted. 5 Write the nswer to the speified level of ury with the orret units. Exmple : Finding the surfe re of omposite solid 6J The red ue is sulpture in New York onsisting of ue with ylindril hole removed from the entre. The side length of the ue is 8.5 m nd the rdius of the ylindril hole is.4 m. Wht is the surfe re of the fe with the ylindril hole, orret to two deiml ples? Wht is the surfe re of the red ue? Inlude only the outer red surfes. Answer orret to two deiml ples. Solution: Required fe onsists of squre with irle removed, so use the formul for squre nd irle. Sustitute the vlues into formul. Evlute. 4 Composite solid hs two fes with ylindril holes nd four fes tht re squres. 5 Write down n expression for surfe re using the re of ll the fes. 6 Evlute. 7 Write nswer in the orret units. A s πr 8.5 π.4 66.46 66.4 m SA ( 66.4) + (4 8.5 ) 4.895 4.90 m Surfe re is 4.90 m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6J Surfe re nd volume of omposite solids 5 Volume of omposite solids To lulte the volume of most ommon solids we use formul. Some of these formule re listed erlier in this hpter. Composite solids re mde up of two or more ommon solids. The volume of omposite solid is lulted y dding or sutrting the volumes of eh ommon solid. Volume of omposite solids Divide the omposite solid into two or more ommon solids. Clulte the volume of eh ommon solid. Add or sutrt the volumes of the ommon solids. 4 Write the nswer to the speified level of ury with the orret units. Exmple 4: Finding the volume of omposite solid 6J A grin silo onsists of ylinder with one on top, s shown. The dimeter of the ylinder is 6 m nd its height is 0 m. The dimeter of the se of the one is the sme s tht of the ylinder, nd its height is m. Wht is the volume of the silo orret to one deiml ple? Solution: Use the volume formul for one V πr h. Sustitute r 8 nd h into the formul. Evlute. 4 Use the volume formul for ylinder V πr h. 5 Sustitute r 8 nd h 0 into the formul. 6 Evlute. 7 Add the volume of the one to the volume of the ylinder. 8 Evlute. 9 Write the nswer orret to one deiml ple nd with orret units. V V V πr h (Cone) π 8 4.04866 m πr h(cylinder) π 8 0 40.8597 m Cone + Cylinder 4.04 + 40.8 455. m Volume of silo is 455. m. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6 Chpter 6 Perimeter, re nd volume 6J Exerise 6J LEVEL Eh of the following solids is mde up of m ues. Wht is the surfe re nd volume of eh solid? Wht is the surfe re nd volume of eh solid? Answer to the nerest whole numer. 4 m 9 m m 5 m m m 8 m 7 m 6 m 9 m 4 m 6 m 0 m 4 m 45 m Wht is the surfe re nd volume of eh solid? Answer to the nerest whole numer. 8 mm 4 mm m 6.7 m 4 m.8 m 4 A hemisphere of ie rem is pled on wffle one. The wffle one hs rdius of.5 m, perpendiulr height of m nd slnt height of.5 m. Answer the following questions orret to one deiml ple. Wht is the (outside) surfe re of the wffle one? Wht is the surfe re of the hemisphere of ie rem? Wht is the totl surfe re of the wffle one with the ie rem? d Wht is the volume of the wffle one? e Wht is the volume of the hemisphere of ie rem? Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6J Surfe re nd volume of omposite solids 7 Exmple 4 LEVEL 5 Wht is the surfe re nd volume of eh solid? Answer to the nerest whole numer. m m 5 m 4 m 0 m 5 m m 4 m 7 m m 6 A one sits extly inside ylinder s shown. The height of the one is 6 m. Wht is the volume of the ylinder to the nerest ui entimetre? Wht is the volume of the ylinder not tken up y the one? Answer to the nerest ui entimetre. 8 m 7 A omposite solid onsists of two squre pyrmids with their ses touhing, s shown. The side length of the se is 40 m nd the height of the solid is 80 m. Find the volume of the solid orret to the nerest ui entimetre. 40 m 80 m LEVEL 8 A trpezoidl prism hs ylindril hole of rdius of 0.5 m ut out s shown. Wht is the volume of the ylindril hole? Answer orret to two deiml ples. Wht is the re of the trpezoidl se or front fe? Wht is the volume of the trpezoidl prism fter the hole is ut? Answer orret to two deiml ples. m m m 4 m 9 The first of two hot wter tnks is squre prism with side length of 80 entimetres nd height of.9 metres. The seond hot wter tnk is ylinder with dimeter of 80 entimetres nd height of.9 metres. Wht is the volume of eh tnk? Answer in ui metres orret to three deiml ples. Wht is the surfe re of eh tnk? Answer to the nerest squre metre. 0 A squre pyrmid hs side length of 00 m nd perpendiulr height of 80 m. A frustum is formed y removing the top of the squre pyrmid with side length of 00 m. Find the volume of the frustum to the nerest ui entimetre. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
8 Chpter 6 Perimeter, re nd volume 6K 6K Cpity The pity of ontiner is the mount of liquid it n hold. Some solids hve oth volume nd pity. For exmple, n of soft drink is ylinder tht hs volume ( V πr h ) nd pity (60 ml). The se unit for pity is the litre (L). Three ommonly used units for pity re the meglitre (ML), kilolitre (kl) nd millilitre (ml). ML 000 kl ML000000 L kl000 L L 000 ml Cpity m ml m 0.00 L 000 m L m m m m 000000 m 000000 ml 000 L kl Exmple 5: Finding the pity 6K The ontiner shown is filled with wter. Find the volume of the ontiner in ui entimetres. Find the pity of the ontiner in litres. 0 m Solution: Use the volume formul for right prism V Ah. Determine the shpe of the se nd the formul to lulte the re of the se A l. Sustitute the vlues into the formul. 4 Evlute. 5 Give nswer to the orret units. 6 To hnge m to L multiply y 0.00 ( m 0.00 L). 7 Alterntive method is to onvert to ml ( m ml). V Ah 70 m lh 70 40 0 84000 m Cpity 84 000 0.00 L 84 L Cpity 84000 ml 84000 ml 84 L 40 m Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
6K Cpity 9 Exerise 6K LEVEL A n of soft drink hs pity of 75 ml. How mny ns of soft drink would it tke to fill. L ottle? How muh would remin? A mediine ottle hs pity of 0. L. Wht is the pity in millilitres? How mny tlespoons (5mL) does the ottle ontin? How mny tespoons (5 ml) does the ottle ontin? d The orret dosge is 0 ml, times dy. How mny doses does the ottle ontin? Complete the following. 4m ml 000 m L 70 m ml d 4000 m L e 900 m ml f 500 m L g 4 m kl h 0 m L i 0 m kl j 7m L k 5m kl l 8m ml Exmple 5 4 Wht is the pity of retngulr prism whose se re is 0 m nd height is 0 m? Answer orret to the nerest millilitre. 5 Find the pity of tringulr prism with height of 8 m nd se re of 40 m. Answer in litres, orret to two signifint figures. 6 Find the pity of retngulr pyrmid whose se re is m nd height is 5 m. Answer orret to the nerest millilitre. 7 Find the pity of ylindril plsti ontiner 6 m high nd with irulr end surfes of rdius 8m. Answer orret to the nerest litre. 8 Find the pity of ue whose side length is 75 mm. Answer in millilitres, orret to two deiml ples. Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press
40 Chpter 6 Perimeter, re nd volume 6K 9 Find the pity of the following solids in millilitres, orret to two deiml ples. LEVEL 8 m m 4 m 5 mm 4 mm 6 mm mm 8 mm 0 mm d e A m f 0 m 9 m 7 m 6 m 0 m m 0 A wter tnk is the shpe of ylinder with rdius of m nd height of.5 m. Wht is the re of the top irulr fe of the wter tnk? Answer orret to one deiml ple. Determine the volume of the wter tnk in ui metres. Answer orret to one deiml ple. Wht is the pity of the tnk, to the nerest kilolitre? m.5 m LEVEL A swimming pool is the shpe of retngulr prism s shown. The swimming pool is filled to 5 m from the top. Wht is the volume of wter in ui metres? How muh wter does the swimming pool ontin, to the nerest kilolitre? 5 m 0 m.7 m A swimming pool is the shpe of 8.8 m 0.5 m trpezoidl prism s shown opposite.. m Find the volume of the pool in.5 m ui metres. Wht is the pity of the pool in kilolitres? In hot wether the level of the pool dropped y 75 m. The pool ws originlly full. Wht volume of wter, in litres, evported? Mthemtis Stndrd Yer ISBN 978--08-446-8 Powers 08 Cmridge University Press