9 LENGTH, AREA AND VOLUME


 Austen Nicholson
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1 E 9E Are of prllelogrm 9B Perimeter 9F Are of tringle M PL 9A Unerstning length 9C Unerstning re 9G Surfe re 9D Are of retngle 9H Volume n pity SA MEASUREMENT AND GEOMETRY 9 LENGTH, AREA AND VOLUME E SS E N TIAL QUE S TION Wht is the sme n wht is ifferent out mesuring twoimensionl shpes n threeimensionl ojets? _NOL_MM7_TXT_SI.in /07/14 11:51 AM
2 Are you rey? 9A 1 Use the sle on the ruler to mesure the length of the line. MM CM E 5 Whih of these shpes is prllelogrm? A B C D 6 Wht is the length of this line? B 29 mm C 34 mm D 39 mm 9E prllelogrm is not true? A A prllelogrm hs four sies. 2 To tre roun this shpe, wht B Opposite sies hve equl length. istne woul the tip of your penil trvel? C Opposite sies re prllel. 1 m A 6 m C 10 m D 12 m 9C D All sies re ifferent lengths. 2 m M PL B 8 m 6 Whih of these sttements out E 9B A 24 mm 4 m 9F 7 How mny sies oes tringle hve? Questions 8 n 9 refer to this ojet. 5 m 3 Wht is the re of this shpe? 9G 8 How mny fes oes this ojet hve? 9G 9 Whih fe is the sme size n shpe s SA the front fe? 9D A 6 squre units B 7 squre units C 8 squre units D 9 squre units 9H A k fe B top fe C ottom fe D right fe 10 This ojet is me up of loks. Eh lok is 1 m long, 1 m wie n 1 m high. Choose the sttement tht orretly esries the imensions of this ojet. 4 This retngle is rwn on 1m gri pper. Wht is its re? A length 4 m, with 5 m, height 1 m A 5 m2 B 8 m2 C 15 m2 D 16 m2 B length 4 m, with 3 m, height 2 m C length 2 m, with 3 m, height 2 m D length 4 m, with 4 m, height 3 m _NOL_MM7_TXT_SI.in /07/14 11:51 AM
3 464 CHAPTER 9: LENGTH, AREA AND VOLUME 9A Unerstning length Strt thinking! Length n e mesure in millimetres (mm), entimetres (m), metres (m) or kilometres (km). 1 Look roun the room. Fin n ojet tht hs length imension (length, with or height) of 1 mm. 2 Look for n ojet tht hs length of 1 m n nother tht hs length of 1 m. Visulising these unit lengths will help you estimte the length of things. 3 Look t your ruler. How mny millimetres re equivlent to 1 m? 4 Look t tpe mesure or metre ruler. How mny entimetres re equivlent to 1 m? How mny millimetres re equivlent to 1 m? 5 How mny metres re equivlent to 1 km? (Think out the prefix kilo. Wht oes kilo men?) 6 Complete these sttements. mm = 1 m m = 1 m m = 1 km 7 Disuss with prtner how to onvert length mesurement from entimetres to millimetres. How is this ifferent from onverting millimetres to entimetres? Woul this metho work for other units? 8 Explin how to onvert length mesurement to: smller unit lrger unit. KEY IDEAS Metri units of length re millimetres (mm), entimetres (m), metres (m) n kilometres (km). If you strt y estimting length, you n eie whether your length mesurement is resonle. A unit of length n e onverte to nother unit of length y multiplying or iviing y the onversion ftor Multiply y the onversion ftor to onvert to smller unit. km m m mm Divie y the onversion ftor to onvert to lrger unit
4 9A UNDERSTANDING LENGTH 465 EXERCISE 9A Unerstning length UNDERSTANDING AND FLUENCY EXAMPLE 9A1 1 Estimte the length of eh line segment. 2 Use ruler to urtely mesure the length of eh line segment in question 1. 3 Complete eh sttement y using the wors multiply or ivie. When onverting length mesurement to smller unit, you nee to y the onversion ftor. When onverting length mesurement to lrger unit, you nee to y the onversion ftor. Complete these length onversions. 5.4 km = m 900 mm = m THINK Converting units of length Multiply y 1000 s you re onverting to smller unit. (1 km = 1000 m.) Divie y 10 s you re onverting to lrger unit. (10 mm = 1 m.) WRITE 5.4 km = ( ) m = 5400 m 900 mm = (900 10) m = 90 m 4 Complete these length sttements. 8 km = m 3 m = m 700 mm = m 500 m = m e 16 m = mm f 2000 m = km g 3.5 m = m h 9 mm = m i 1.34 km = m j 0.47 m = mm k 84 m = km l 51.2 m = m m 0.06 km = m n 3.8 m = m o 7.9 mm = m p 4310 mm = km
5 466 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY 5 Complete these length sttements. 4 km = m = m mm = m = m 73 m = m = mm 8400 m = m = km e mm = 250 m = m f m = 362 m = km 6 Convert these mesurements to the units shown in rkets. 55 m (mm) m (km) 41 mm (m) 6.2 m (m) e 3.6 km (m) f 89 m (m) g 1.7 km (m) h 500 mm (m) i 7000 m (km) j 3.65 m (mm) k 0.94 km (mm) l 1800 mm (km) 7 Some stuents me errors when they were onverting some length mesurements. Explin the error in eh se n give the orret nswer. 55 m = 5.5 mm 24 km = 2400 m 0.66 m = 60.6 m 1.73 mm = m e 420 m = 42 m f 800 m = 0.08 km 8 Put the length mesurements in eh list into sening orer (from smllest to lrgest). 5 m, 15 mm, 60 mm 2 m, 190 m, 210 mm 0.7 km, 701 m, 7001 m 82 mm, 8 m, 0.8 m e 3 km, 3000 mm, 300 m, 30 m f 5500 mm, km, 5.7 m, 560 m 9 Without using ruler, rw line segment 7 m long. Hve lssmte use ruler to hek how lose your line segment is to 7 m. 10 Give three exmples where the length woul e est mesure in eh of these units. mm m m km 11 Without using ruler, estimte the length of the terpillr in: i entimetres ii millimetres. Compre your nswers to prt with lssmte. Disuss how you eh got your nswers. Use ruler to urtely mesure the length of the terpillr. Think of some resons why you n your lssmte my not get the sme mesurement.
6 9 A U N D E R S TA N D I N G L E N G T H 467 inite. E U N DE R S TA N DING A N D F LU E NC Y 12 Here re some lifesize photos of ojets. First estimte n then mesure the lengths 13 Without using ruler, estimte eh of these. height of your esk M PL length of your pen with of your fingernil height of the oor e with of the plm of your hn f length of the room 14 Disuss your nswers to question 13 with lssmte. Do the mesurements soun resonle? Use ruler or tpe mesure to hek how lose your estimtes re to the tul mesurements. 15 Whih unit of length (mm, m, m or km) woul e most suitle for mesuring eh of these? length of relet height of tennis net thikness of CD e epth of fish pon f P R O B L E M S O LV I N G A N D R E A S O N I N G SA istne to the Moon 09_NOL_MM7_TXT_SI.in 467 length of riket pith 16 The length of swimming pool is 50 m. How fr woul you swim if you omplete 25 lps? Woul this istne e more or less thn 1 km? How mny lps woul you swim to omplete istne of 2 km? 17 A rem of pper ontins 500 sheets. Clulte the thikness of single sheet of pper. Stte your nswer in millimetres. 18 A group of people re lining up to uy tikets for onert. Estimte how long the queue will e if there re 85 people in the line. Explin how you rehe your nswer. 5.2 m 15/07/14 11:51 AM
7 CHAPTER 9: LENGTH, AREA AND VOLUME P R O B L E M S O LV I N G A N D R E A S O N I N G 19 Estimte the following length mesurements using the informtion supplie in eh photo. height of the ult girffe (m) height of the pelin (m) 50 m 360 m length of the snke (m) M PL length of the mouse s til (mm) E 468 SA 17 m 20 Different instruments n e use to mesure length; l th for f exmple, l 20 m ruler, tpe mesure or trunle wheel. Choose something in your lssroom suh s esk, tle, hir or the floor. Use n pproprite instrument to tke length mesurements. Drw igrm n show ll the mesurements you foun. Rememer to inlue the unit you use. 09_NOL_MM7_TXT_SI.in /07/14 11:51 AM
8 9A UNDERSTANDING LENGTH 469 Choose something outsie in the shool grouns tht oul e mesure in metres or perhps kilometres. Agin, mesure it using n pproprite instrument. Drw igrm n show ll the mesurements you foun, inluing the units. Show your results for prts n to lssmte. Explin your hoie of mesuring instrument n length unit use. Were these the est hoies? CHALLENGE 21 Estimte the length of this onneting or, exluing the onnetors. Desrie how you oul urtely mesure the length of the or from the photo. Wht equipment oul you use? Fin the length of the or. Compre your result to the estimte you me in prt. 22 Estimte the length of eh of these. Then use string or otton n ruler to mesure their length. e Reflet How o you eie whih instrument n unit to use when mesuring length?
9 470 CHAPTER 9: LENGTH, AREA AND VOLUME 9B Perimeter Strt thinking! The ourse for hrity wlk is shown. The wlkers strt n finish t the sme ple. 3.4 km hekpoint 1 hekpoint 2 E 1.3 km hekpointt km 1.9 km M PL strt/finish 3.1 km hekpoint 4 1 Wht is the istne from the strt to the: first hekpoint? seon hekpoint? nt? thir hekpoint? fourth hekpoint? t? 2 Clulte the totl istne roun the ourse from strt to finish. Explin how you worke this out. 3 Compre your nswer to question 2 with tht of some lssmtes. Explin why they my e ifferent. SA 4 The totl istne roun the ourse from strt to finish is lso lle the perimeter of the ourse. Why re the istnes you lulte in question 1 not onsiere to e the perimeter? KEY IDEAS Perimeter is the istne roun the outsie ege of twoimensionl (2D) shpe. Sine perimeter is length mesurement, ommon units re millimetres, entimetres, metres n kilometres. To lulte the perimeter, the length of eh sie together. All length mesurements must e in the sme unit efore eing e. 09_NOL_MM7_TXT_SI.in /07/14 11:51 AM
10 9B PERIMETER 471 EXERCISE 9B Perimeter EXAMPLE 9B1 Clulting perimeter Clulte the perimeter of this shpe. 2.4 m 1.7 m UNDERSTANDING AND FLUENCY THINK 1 There re three sies to the shpe so there re three lengths to e e. Write the lultion. 2 Chek whether ll the mesurements re in the sme unit (yes). 3 A ll the lengths together n inlue the pproprite unit. 1 Clulte the perimeter of eh shpe. e 6 km 4 m 10.2 mm 5 m 9 km 8 km 8 m 7.1 mm 6 m 3.6 mm 4.5 mm 3.5 m WRITE perimeter = 1.7 m m m f 5 m 7 mm 9 mm 16 m = 7.6 m 14 m 19 m 7 m 7 m 8 mm 10 mm 37 m 5 m 11 mm 35 m 7.4 mm 23 m
11 472 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY 2 Clulte the perimeter of these shpes, whih re shown on 1m gri pper. 3 Clulte the perimeter of eh shpe y using ruler to mesure the lengths of the sies. 4 Fin the perimeter of eh shpe y using ruler to mesure eh sie to the nerest millimetre.
12 9B PERIMETER 473 EXAMPLE 9B2 Clulting perimeter fter onverting units Clulte the perimeter of this shpe in entimetres. 22 m 13 m 110 mm 50 mm 17 m UNDERSTANDING AND FLUENCY THINK 1 There re five sies to the shpe so there re five lengths to e e. Write the lultion. 2 Chek whether ll the mesurements re shown in the sme unit (no). Express them in entimetres. 3 A ll the lengths together n inlue the pproprite unit. 5 Clulte the perimeter of eh shpe in entimetres. e 2 m 50 mm 5.3 m 5.3 m 70 mm 4 m 36 mm 36 mm 0.3 m 68 m 74 m 1.2 m WRITE perimeter = 22 m + 13 m + 50 mm + 17 m mm f 0.6 m 75 m 75 m 1.5 m 25 mm 90 m 25 mm 15 mm 15 mm 5.1 m = 22 m + 13 m + 5 m + 17 m + 11 m = 68 m 34 mm 4.6 m 20 mm 2.7 m 1.3 m 15 mm 18 mm
13 474 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY 6 Fin the lengths of ll the sies n then lulte the perimeter of eh shpe. 4 m 10 m 8 m 2 mm 8 mm 5 mm 2 mm 3 mm 3 m 9 m 2 mm 5 m 3 m 4 m 1.3 m 2.5 m 1.5 m 1.9 m 7 m 4.2 m 7 A retngle hs length 11 m n with 4 m. The sies 11 m of this retngle re mrke with either or symols. Explin wht these symols men. 4 m Clulte the perimeter of this retngle. Inste of ing the four sies, is there quiker wy to lulte the perimeter of retngle? Desrie this shortut. If l is use to represent the length of retngle n w is use to represent the with of retngle, write formul for fining the perimeter of retngle. 8 Use the formul to fin the perimeter of these retngles. length 5 m, with 3 m length 12 m, with 10 m length 47 mm, with 35 mm length 100 m, with 24 m e length 6 m, with 4.5 m f length 3.2 m, with 1.9 m g length 8.62 m, with 425 m h length 15 mm, with 1.2 m 9 Clulte the perimeter of eh shpe. Look for shortut to use in eh se. 3 m 11 mm 2.5 km 8 m 10 For eh shpe in question 9, write formul for the perimeter using l to represent the length of sie.
14 475 9B PERIMETER thir se n finlly home. Use the mesurements shown to nswer the following. How fr oes tter run to reh first se fter hitting the ll? How fr oes tter run to reh thir se fter hitting the ll? How fr oes tter run to reh the home plte when soring home run? m m thir se first se 27.4 m home plte 4m 27. tting plte M PL Whih one out of prts, n is the sme s the perimeter of the sell imon? seon se E P R O B L E M S O LV I N G A N D R E A S O N I N G 11 In sell, tter must run from the tting plte to first se, then seon se, 12 A volleyll ourt is 18 m long n 9 m wie. Drw igrm of the ourt showing its mesurements n the net. Clulte its perimeter. To wrm up, eh tem jogs five lps roun their hlf of the ourt. Tht is, the plyers jog long the outsie lines n long the net on their sie of the ourt. How fr oes plyer jog uring the wrmup? 13 The photo shown hs perimeter of 52 m. SA Wht is the with of the photo? 14 A squre rink oster hs perimeter of 342 mm. Wht is the length of the rink oster? 17 m 15 A grener plns to ig retngulr her gren with orer 36 m long. List five sets of possile imensions for the her gren. 16 A Hills hoist lothesline hs four lines rrnge CHALLENGE in squres. Use the mesurements shown on the igrm (looking t the lothesline from ove) to lulte the totl length of line use. 2.4 m 24 m 24 m 24 m 17 A retngulr swimming pool of length 7.5 m n with 4.6 m is to hve sfety fene uilt roun it. If the fene will e 2 m from the ege of the pool, lulte the totl length of fening neee (inlue the gte). 09_NOL_MM7_TXT_SI.in 475 Reflet Explin to frien how to lulte the perimeter of shpe. 15/07/14 11:51 AM
15 476 CHAPTER 9: LENGTH, AREA AND VOLUME 9C Unerstning re Strt thinking! re of 1 m 2 The mount of spe enlose y the outsie ege of 2D shpe is lle the re. Common units use for re re squre millimetres (mm 2 ), squre entimetres (m 2 ), squre metres (m 2 ) n squre kilometres (km 2 ). 1 Preit whih item hs the lrger re: the postge stmp or the 50ent oin. 2 The outline of the stmp is tre on to 1m gri pper. Wht is the re of the stmp in squre entimetres? How i you work this out? 3 The outline of 50ent oin is tre on to 1m gri pper. Wht is the re of the oin in squre entimetres? How i you work this out? 4 Di you fin n ext or pproximte vlue for the re of the oin? Give reson. 5 Whih hs the lrger re: the stmp or the oin? Ws your preition orret? KEY IDEAS Are is the mount of spe enlose y 2D shpe. Common units of re re squre millimetres (mm 2 ), squre entimetres (m 2 ), squre metres (m 2 ) n squre kilometres (km 2 ). The re of shpe n e foun y ounting the numer of whole squre units overe y the shpe s well s ny hlf squre units overe. A squre with sie length 1 m hs re 1 m 2. For n irregulr shpe, the re n e pproximte y ounting ll the squre units tht re overe n ll the squre units tht re t lest hlf overe y the shpe. this squre is ounte this squre is not ounte
16 9C UNDERSTANDING AREA 477 EXERCISE 9C Unerstning re EXAMPLE 9C1 Fining re y ounting squres UNDERSTANDING AND FLUENCY Fin the ext re of this shpe rwn on 1m gri pper. THINK 1 Count the numer of whole squres n hlf squres within the shpe. 2 Write the re of the shpe. Eh squre in the gri hs n re of 1 m 2. 1 Fin the ext re of these shpes rwn on 1m gri pper. WRITE 3 whole squres + 2 hlf squres = 4 whole squres re = 4 m 2 e f
17 478 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY 2 These shpes re shown on 1m gri pper. i iii EXAMPLE 9C2 Fin the re of eh shpe. Fin the perimeter of eh shpe. Explin the ifferene etween fining the re n fining the perimeter of eh shpe. Estimte the re of this shpe rwn on 1m gri pper. THINK Estimting re 1 Some squres re not totlly enlose in the shpe. Count the squres tht re ompletely enlose n those tht re t lest hlf overe y the shpe. Tik the squres to e ounte. 2 Count the numer of squres tht re tike n write your nswer for the re. To show your nswer is pproximte, use the symol, whih mens is pproximtely equl to. WRITE ii iv re 10 m 2
18 9C UNDERSTANDING AREA 479 UNDERSTANDING AND FLUENCY 3 Estimte the re of eh shpe rwn on 1m gri pper. 4 Work with prtner to nswer these questions. i Drw squre tht hs n re of 1 mm 2. ii Give two exmples where re woul est e mesure in mm 2. i Drw squre tht hs n re of 1 m 2. ii Give two exmples where re woul est e mesure in m 2. i Use some sheets of newspper, sissors n stiky tpe to mke squre tht hs n re of 1 m 2. ii Give two exmples of situtions where re woul est e mesure in m 2. i Desrie the size of 1 km 2. ii Give two exmples of situtions where re woul est e mesure in km 2. 5 Use your nswers to question 4 to estimte the re of these. top of your esk front over of your textook one wll of your lssroom surfe of your thum nil e sole of your shoe f floor of your lssroom
19 480 CHAPTER 9: LENGTH, AREA AND VOLUME ple over eh photo. lef oil spot on grge floor M PL E P R O B L E M S O LV I N G A N D R E A S O N I N G 6 Fin the re overe y eh of these (shown tul size). A 1m gri hs een SA og pwprints oin 7 Whih unit of re (mm2, m2, m2 or km2) woul e est for mesuring the re of these? sketll ourt wristwth fe ntionl prk fiveollr note e surfe of skteor rmp f fmilysize pizz 8 Another unit of re is the hetre (h). This unit is often use 100 m for ln re. One hetre hs the re of squre with sie lengths of 100 m. Tht is, 1 h = m2 or 100 h = 1 km2. A iry frm overs n re of 8 km2. Wht is the re of the frm in hetres? 1 hetre 100 m A 40h property is vertise for sle. Wht is the ln re of the property in m2? 09_NOL_MM7_TXT_SI.in /07/14 11:51 AM
20 9 C U N D E R S TA N D I N G A R E A 481 CHALLENGE 9 A gri hs een ple over mp of Austrli where eh squre in the gri overs km2. Fin the pproximte re of Austrli in km2. M PL The offiil re of Austrli (inluing islns) is km2. Compre your pproximtion with this. Explin why the vlues re ifferent n suggest how you oul improve the ury of your pproximtion. E Express the re of Austrli in hetres. 10 Explin why there re situtions where you n fin the ext re n others where you n only fin the pproximte re. Provie exmples to support your nswer. SA 11 Use the 1m gri to estimte the re of this rtoon hrter. Reflet Wht is the ifferene etween mesuring length n mesuring re? 09_NOL_MM7_TXT_SI.in /07/14 11:51 AM
21 482 CHAPTER 9: LENGTH, AREA AND VOLUME 9D Are of retngle Strt thinking! 1 Using 1m gri pper, rw three ifferent retngles. In eh se mke the mesurements for the length n with whole numer of entimetres. 2 Copy n omplete this tle. Retngle Length (m) With (m) Are (m 2 ) Desrie shortut you oul use to lulte the re of eh retngle without ounting the squres in the gri. 4 Use the wors length, with n re to write rule for lulting the re of retngle. 5 When woul this rule e more useful thn ounting the squres in gri? KEY IDEAS Rule for re of retngle: Are = length with Inste of using wors, pronumerls n e use to write formul for lulting the re of ny retngle. Formul for re (A) of retngle, where l represents the length mesurement n w represents the with mesurement: A = l w (or A = lw) l w The mesurements for l n w must e in the sme unit efore sustituting them into the formul.
22 9D AREA OF A RECTANGLE 483 EXERCISE 9D Are of retngle EXAMPLE 9D1 Clulting re of retngle using formul UNDERSTANDING AND FLUENCY Clulte the re of this retngle. THINK 1 Clulte the re of eh retngle. 8 m 40 mm 5 m 11 m 1 Ientify the shpe (retngle). Write the formul for its re. 2 Ientify the mesurements for l n w. Chek they re in the sme unit (yes). 10 mm 6 m WRITE A = l w l = 11 m, w = 6 m 3 Sustitute the vlues for l n w into the formul. A = 11 m 6 m 4 Clulte the result n inlue the pproprite unit. = 66 m 2 16 m 3 m 2 m 7 m e 28 mm f 2.1 m 19 mm 4.3 m
23 484 CHAPTER 9: LENGTH, AREA AND VOLUME 2 Fin the re of the retngles with these imensions. length 9 m, with 7 m length 13 mm, with 4 mm length 6 m, with 1 m length 10.4 m, with 5 m e length 14.2 mm, with 9.7 mm f length m, with 5.4 m EXAMPLE 9D2 Clulting re of retngle fter onverting units UNDERSTANDING AND FLUENCY Clulte the re of this retngle. THINK 58 mm 1 Ientify the shpe (retngle). Write the formul for its re. 2 Ientify the mesurements for l n w. Chek tht they re in the sme units (no). Convert l to entimetres so tht oth l n w re in the sme units. 2.5 m WRITE A = l w l = 58 mm = 5.8 m w = 2.5 m 3 Sustitute the vlues for l n w into the formul. A = 5.8 m 2.5 m 4 Clulte the result n inlue the pproprite unit. = 14.5 m 2 NOTE w oul hve een onverte to millimetres n the re foun to e 1450 mm 2. 3 Clulte the re of eh retngle. 50 mm 4.5 m 2 m 600 m e 0.2 m f 32 mm 0.6 km 50 mm 140 m 175 mm 5 m 1.8 m 4 Fin the re of eh retngle with these imensions. length 3 m, with 100 m length 8 m, with 50 mm length 4 km, with 280 m length 600 mm, with 30 m e length 3.1 m, with 20 mm f length 8.4 m, with 0.05 m
24 9D AREA OF A RECTANGLE 485 UNDERSTANDING AND FLUENCY 5 Clulte the re of eh squre. 4 m 5 m 17 mm 3.6 m 4 m 6 Write formul for the re of squre of length l. Use your working from question 5 to help you. 7 Fin the re of the squres with these sie lengths. 7 m 15 m 11 km 60 mm e 6.4 m f 180 m g 0.7 mm h 0.53 km 8 The shpe shown in Figure A n e ivie into two retngles (Figure B). Figure A 6 m 5 m 8 m 4 m 3 m 2 m Figure B 6 m 5 m re 1 8 m Clulte the re of the retngle lelle s re 1. Clulte the re of the retngle lelle s re 2. Wht is the totl re of the shpe? Is there nother wy of iviing the shpe into two retngles? Drw igrm to show how this oul e one. e Clulte the re of the shpe using your igrm from prt. 3 m re 2 9 Copy eh igrm n show how eh shpe n e ivie into two or more retngles. 11 mm 4 m 2 m 6 mm 1 m 3 m 9 mm 5 m 4 mm 3 mm 8 m 2 m 7 mm 6 mm 5 m 3 mm 7 m 5 m 9 m 4 m 5 m 12 m 8 mm 4 mm 15 m
25 486 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY 10 Fin the re of eh shpe in question In questions 9 n 10, the shpes re ivie into seprte retngles n the re of eh retngle e together to fin the totl re of the shpe. Explin how you oul lulte the re of the she region in this shpe using sutrtion inste. Wht is the re? 12 Fin the re of the she region in eh of these. 13 m 14 mm 5 mm 3 m 5 mm 2 mm 5 m 10 m 2 mm 7 m 6 m 2 m 11 m 5 m 13 Fin i the re n ii the perimeter of eh shpe. 9 m 3 m 7 m 2.9 m 4 m 10.7 m 4.9 m 2.9 m 4.7 m 5.4 m 1.8 m 5 m 3 m 1 m 1 m 3.1 m 6 mm 7 mm 10 mm 2.7 m 7 mm 7 mm 10 m 3 m 3 m 6 m 6 mm 8 mm 4.3 m 14 Write three sets of possile imensions for retngles with these res. 48 m 2 15 m 2 15 How mny sets of possile imensions re there for squre tht hs n re of 16 mm 2? List the possiilities.
26 9D AREA OF A RECTANGLE 487 seline PROBLEM SOLVING AND REASONING CHALLENGE 16 A tennis ourt hs plying re tht is ivie y lines rwn on the groun. Use the imensions shown to lulte eh of these. i the length from the seline to the net ii the with of the oules plying re (the full ourt) iii the with of the singles plying re (the outer sielines re not use in singles mthes) iv the plying re use in oules mth v the plying re use in singles mth m net smple serving re vi the re of the ourt tht plyer n serve the tennis ll into 8.23 m 6.4 m seline m When it strts to rin, wterproof over is ple over the plying surfe. Suggest how over or overs oul e esigne to e quikly n effetively use. Drw igrm with the imensions lelle to show how the ourt oul e overe. Wht mount of wterproof mteril woul e require for the over? 17 Wht is the with of the sreen of this iphone (shown t ner right), if its re is 3750 mm 2? 18 If the re of the sreen on the BlkBerry phone (shown t fr right) is 3000 mm 2, wht is its length? 19 The perimeter of retngle with length of 14 m is 46 m. Wht is its re? 75 mm 20 The perimeter of squre is 80 m. Wht is its re? 21 The perimeter of retngle is 64 m. Wht is its re? Explin how you worke out your nswer. 22 To mrk out the ounry of new vegetle gren, I hve een given piee of string 12 m long. List the imensions of five possile retngulr vegetle grens I oul mke. Whih of these five woul hve the lrgest re? 5 m 23 Does retngle exist where the perimeter hs the sme vlue (mesure in m) s the re (mesure in m 2 )? Explin your nswer. Reflet How woul you explin to frien the ifferene etween the re n the perimeter of retngle?
27 488 CHAPTER 9: LENGTH, AREA AND VOLUME 9E Are of prllelogrm Strt thinking! A prllelogrm is foursie shpe with two pirs of prllel sies n opposite sies equl in length. The imensions se n height re lwys t right ngles to eh other. KEY IDEAS height height se se 1 Copy this prllelogrm on to sheet of pper n refully ut roun the shpe. 2 Cut out the she tringle n ple it t the opposite en of the prllelogrm to form retngle. 3 Drw igrm to show how you hnge the prllelogrm into retngle. Lel the igrm with se n height. 4 Explin how you n use the imensions of the retngle to work out the re of the prllelogrm. Rule for re of prllelogrm: Are = se height height se se height Formul for re (A) of prllelogrm, where represents the se length n h represents the height: A = h (or A = h) The se n height re perpeniulr (t right ngles) to eh other. h The mesurements for n h must e in the sme unit efore sustituting into the formul.
28 9E AREA OF A PARALLELOGRAM 489 EXERCISE 9E Are of prllelogrm UNDERSTANDING AND FLUENCY EXAMPLE 9E1 1 Copy eh prllelogrm n lel the se n the height. 6 m 8 mm e 4 m 18 m 7 mm 10 m 20 m 16 m 12 m 9 mm f 2.4 m 5 mm 4 mm 1.5 m Clulte the re of this prllelogrm. THINK 4 mm 7 mm 1 Ientify the shpe (prllelogrm). Write the formul for the re of prllelogrm. 2 Ientify the mesurements for n h. Chek tht they re in the sme unit (yes). 2.9 m Clulting re of prllelogrm using formul WRITE A = h = 7 mm h = 4 mm 3 Sustitute the vlues for n h into the formul. A = 7 mm 4 mm 4 Clulte the result n inlue the pproprite unit. = 28 mm 2 2 Clulte the re of eh prllelogrm in question 1. 3 Fin the re of eh prllelogrm with these imensions. se length 5 m, height 3 m se length 10 mm, height 17 mm se length 4 m, height 1.5 m se length 2.5 m, height 4.1 m
29 490 CHAPTER 9: LENGTH, AREA AND VOLUME EXAMPLE 9E2 Clulting re of prllelogrm fter onverting units Clulte the re of this prllelogrm in m mm 3.5 m UNDERSTANDING AND FLUENCY THINK 1 Ientify the shpe (prllelogrm). Write the formul for the re of prllelogrm. 2 Ientify the mesurements for n h. Both mesurements nee to e in entimetres so onvert the height mesurement. 4 Clulte the re of eh prllelogrm in m 2. e 50 m 0.8 m 600 m 1.4 m 72 mm 25 mm 2.3 m 48 mm 9 m 8 m 7 m 1.6 m f WRITE A = h = 3.5 m h = 12 mm = 1.2 m 3 Sustitute the vlues for n h into the formul. A = 3.5 m 1.2 m 4 Clulte the result n inlue the pproprite unit. = 4.2 m 2 35 mm 29 mm 5 Fin the re of eh prllelogrm with these imensions. se length 30 mm, height 5 m se length 58 m, height 200 m se length 0.06 m, height 40 mm se length 27.4 m, height 27.4 mm
30 9E AREA OF A PARALLELOGRAM 491 UNDERSTANDING AND FLUENCY 6 Copy eh igrm n show how eh shpe n e ivie into two or more retngles or prllelogrms. 10 m 4 m 14 m 12 m 6 m 4 m 3 m 7 Fin the re of eh shpe shown in question 6. 4 mm 15 mm 7 m 7 mm 4 mm 8 Fin the re of the she region in eh of these. (Consier sutrting res s well s ing res to otin your nswer.) 15 m 11 m 6 m 22 mm 13 mm 3 m 4 m 7 m 11 m 5 mm 17 mm 11.2 m 5.5 m 16.5 m 7.8 m 9 Fin i the re n ii the perimeter of eh shpe. Use ruler to otin the neessry mesurements. 3 mm 11 mm 6 mm
31 492 CHAPTER 9: LENGTH, AREA AND VOLUME P R O B L E M S O LV I N G A N D R E A S O N I N G 10 Use the mesurements shown on this photo of floor tiles to lulte the re of re tile m 27m 13.5 m 11 A stir ril is supporte y five ientil setions of glss pnels. E Use the imensions shown in the photo to lulte the re of glss neee for ll setions. 1.1 m 0.6 m M PL If the glss ws to e 10 m higher in eh pnel, wht re of glss woul e neee? 12 A ge for jket is me y sewing three piees of fri s shown. The entre SA piee is in the shpe of prllelogrm n the other two piees re retngles. 10 m 3 m 6 m 3 m 8 m 6 m Wht re of fri is neee for the prllelogrm? Wht re of fri is neee for eh retngle? Clulte the totl re of fri neee for this ge. Blk or is sewn roun the outer ege of the ge to mke it stn out on the jket. Clulte how muh lk or is neee. 13 Write three sets of possile imensions for prllelogrms with these res. 24 m2 14 mm2 14 The re of prllelogrm is 20 m2. If the height is 4 m, how long is the se? 15 The re of prllelogrm is 36 m2. If the se is 45 mm long, wht is the height? 09_NOL_MM7_TXT_SI.in /07/14 11:51 AM
32 9 E A R E A O F A PA R A L L E L O G R A M 493 Explin how the re of the rhomus oul e lulte. Do you hve enough informtion to lulte the re? Woul the re of the rhomus e lrger or smller thn 100 m2? Explin your nswer. 17 Consier the retngle shown. Clulte its re using the formul for the re of retngle (A = l w). Clulte its re using the formul for the re of prllelogrm (A = h). Explin why you otin the sme nswer in eh se. 15 m 26 m E (omplete in 1610) is me up of just two ifferent 2D shpes. When you wlk over the floor you experiene the optil illusion of viewing threeimensionl (3D) ojets. SA CHALLENGE 18 The mrletile floor of the hurh of Sn Giorgio Mggiore in Venie, Itly M PL P R O B L E M S O LV I N G A N D R E A S O N I N G 16 The perimeter of the rhomus shown is 40 m. Wht re the two shpes use for the tiles? Wht 3D ojet ppers in the photo t right s result of the optil illusion? Use the imensions shown in the photo for question 10 to lulte the re of the three tiles (lk, re, white) tht mke up the illusion of one 3D ojet. 19 While wlking long street, you notie tht the shpe of the green rrow in trffi light is forme using retngle n two prllelogrms. Estimte the re of the green rrow in the trffi light shown elow using the length mesurement provie. 20 m Reflet Explin why there is onnetion etween the re of prllelogrm n the re of retngle. 09_NOL_MM7_TXT_SI.in /07/14 11:51 AM
33 494 CHAPTER 9: LENGTH, AREA AND VOLUME 9F Are of tringle Strt thinking! A tringle hs three sies. The imensions se n height re lwys t right ngles to eh other. KEY IDEAS height se se Rule for re of tringle: Are = 1 2 se height height The re of tringle n e ompre to the re of prllelogrm or retngle (s retngle is type of prllelogrm). 1 Copy eh prllelogrm shown elow. A tringle hs een forme insie eh shpe. tringle 1 2 Lel the se n the height on eh igrm. se tringle 2 tringle 3 3 How oes the re of the tringle ompre to the prllelogrm in eh se? height 4 If the rule for the re of prllelogrm is se height, write rule for the re of tringle. Formul for re (A) of tringle, where represents the se mesurement n h represents the height: A = 1 2 h (or A = 1 2 h) The se n height must e perpeniulr to eh other. h The mesurements for n h must e in the sme unit efore sustituting into the formul.
34 9F AREA OF A TRIANGLE 495 EXERCISE 9F Are of tringle UNDERSTANDING AND FLUENCY EXAMPLE 9F1 1 Copy eh tringle n lel the se n the height. 5 m 8 m 6 mm 4 mm e f 8 m 6 m 7 m 13 mm 5 mm 10 m Clulte the re of this tringle. THINK 1 Ientify the shpe (tringle). Write the formul for the re of tringle. 2 Ientify the mesurements for n h. Chek tht they re in the sme unit (yes). 6 m 12 mm Clulting re of tringle using formul 5 m WRITE A = 1 2 h = 6 m h = 5 m 3.5 m 3 Sustitute the vlues for n h into the formul. A = m 5 m 4 Clulte the result n inlue the pproprite unit. = 15 m 2 2 Clulte the re of eh tringle in question m 5.2 m 2.4 m 15 m 3 Fin the re of the tringles with these imensions. se 12 m, height 9 m se 24 mm, height 4 mm se 5 m, height 3 m se 8.6 m, height 7 m e se 14.2 mm, height 9.7 mm f se m, height 15.4 m
35 496 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY 4 Drw igrms of three ifferent tringles where the se mesurement is 10 m n the height mesurement is 6 m. Clulte the re of eh tringle. 5 Clulte the re of metl require for eh sign. EXAMPLE 9F2 45 m Clulte the re of this tringle. THINK 60 m 95 m 4.8 m 1 Ientify the shpe (tringle). Write the formul for the re of tringle. 2 Ientify the mesurements for n h. Chek tht they re oth in the sme units (no). Convert h to metres. WRITE A = 1 2 h 75 m Clulting re of tringle fter onverting units = 4.8 m h = 95 m = 0.95 m 3 Sustitute the vlues for n h into the formul. A = m 0.95 m 4 Clulte the result n inlue the pproprite unit. = 2.28 m 2 6 Clulte the re of eh tringle. 8 mm 65 m 0.6 m 182 m 1.5 m
36 9F AREA OF A TRIANGLE 497 UNDERSTANDING AND FLUENCY 3.6 m 45 mm 60 mm e 3.4 m f 50 mm 210 m 30 mm 7 Fin the re of eh tringle with these imensions. se 8 m, height 30 mm se 16 mm, height 3 m se 4 m, height 50 m se 310 m, height 10 m e se 0.7 m, height 20 mm f se 2.8 m, height 97 m 8 Clulte the re of metl require for this sign. 9 A stuent lultes the re of the tringle shown t right to e 90 m 2. Is this orret? Explin your nswer. If it is not orret, work out wht the nswer shoul e. Clulte the perimeter of this tringle. 10 Clulte i the re n ii the perimeter of eh tringle. Use ruler to otin the neessry mesurements. 7.8 m 4 m 0.48 m 15 m 12 m 9 m 41 m
37 498 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY PROBLEM SOLVING AND REASONING 11 Clulte the re of eh omposite shpe. 4 m 7 m 5 m 12 m e f 4 mm 20 mm 10 m 9 mm 5 mm 2 mm 9 mm 2 mm 13 m 8 m 6 m 6 mm 16 m 12 Clulte the mount of silloth neee for the two tringulr sils on this ot. 13 Clulte the re she ornge in eh shpe elow. 20 m 10 m 8 m 4 m 12 m 50 m 12 m 3.5 m 25 m 25 m 18 m 48 m 18 m 48 m
38 9F AREA OF A TRIANGLE Write three sets of possile imensions for tringles with these res. 32 m2 40 m2 15 The re of tringle is 20 m2. If the se is 10 m long, wht is the height? 16 If the totl re tht hs een enmelle on these errings is 270 mm2, lulte how wie eh erring is t the top. 18 mm metl neee for eh of these signs. 24 m E M PL CHALLENGE 17 Use your unerstning of how to fin the re of tringle to lulte the re of 30 m 22.5 m 10.4 m 12 m 26 m SA 18 Estimte the totl re of fri neee to mke the sil use y this winsurfer. Reflet Wht is importnt to rememer out the se n height of tringle? 09_NOL_MM7_TXT_SI.in /07/14 11:51 AM
39 500 CHAPTER 9: LENGTH, AREA AND VOLUME 9G Surfe re Strt thinking! The surfe re of 3D ojet is the totl re of its surfe. Drwing net of the ojet helps us to ientify the shpes tht mke up the surfe. For exmple, the retngulr prism n its net n e rwn for this tissue ox. 1 How mny fes oes the retngulr prism hve? 2 Explin how the net of the prism n help you work out the surfe re. KEY IDEAS The surfe re of 3D ojet is the totl re of the outer surfe of the ojet. The surfe re of prism is the sum of the res of eh fe of the prism. A retngulr prism hs six fes. There re three pirs of ientil fes. Drwing the net of prism n help you see ll the fes of the prism n llow you to lulte the surfe re more esily. left sie ottom left sie front k top k ottom right sie right sie front top
40 9G SURFACE AREA 501 EXERCISE 9G Surfe re UNDERSTANDING AND FLUENCY EXAMPLE 9G1 1 A retngulr prism n its net re shown. How mny fes oes this retngulr prism hve? Desrie the shpe of eh fe. Clulte the re of eh fe of the prism. Use your nswer from prt to lulte the surfe re of the prism. 3 m Clulte the surfe re of this retngulr prism. THINK 1 Work out how mny fes the prism hs. You n rw net of the prism to lerly see ll the fes. 2 Fin the re of eh fe efore ing. Chek tht ll six fes hve een onsiere. 3 Sustitute the pproprite mesurements into the formul for the re of retngle n simplify. WRITE 9 m 4 m 4 m k 4 m left ottom front right 8 m 9 m 9 m top 4 m 4 m Clulting surfe re of retngulr prism 4 m 5 m 5 m 3 m 3 m 4 m 4 m 4 m There re six fes. Eh fe is retngulr in shpe. Surfe re = re of front fe + re of k fe + re of top fe + re of ottom fe + re of leftsie fe + re of rightsie fe = 9 m 4 m + 9 m 4 m + 9 m 5 m + 9 m 5 m + 5 m 4 m + 5 m 4 m = 36 m m m m m m 2 = 202 m 2 3 m 8 m
41 502 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY 2 Clulte the surfe re of eh retngulr prism. 6 mm e 10 mm 15 mm 3.5 m 2.2 m 7 mm 8 mm 5.4 m 3 mm 3 A ue n its net re shown. How mny fes oes this ue hve? Desrie the shpe of eh fe. 3 m f 3 m 3 m 7 m 4 m 6 m 21 m 3 m 11 m 4 m 5 m 9 m 6 m 3 m 3 m 3 m 3 m Clulte the re of eh fe of the ue. 3 m Use your nswer from prt to lulte the surfe re of the ue. e Is there shortut tht n e use to lulte the surfe re of ue? Explin your nswer. 4 Clulte the surfe re of eh ue. 2 m 2 m 2 m 7 mm 7 mm 10 m 7 mm 8 mm 3 m 10 m 10 m 8 mm 8 mm e f 1.5 m 30 m 1.5 m 1.5 m 30 m 30 m
42 9G SURFACE AREA 503 UNDERSTANDING AND FLUENCY 5 A tringulr prism n its net re shown elow. 5 m 5 m 4 m 5 m 5 m 5 m 5 m 4 m 6 m 5 m 9 m 5 m How mny fes oes this tringulr prism hve? Desrie the shpe of eh fe. Clulte the re of eh fe of the tringulr prism. Use your nswer from prt to lulte the surfe re of the tringulr prism. 6 Clulte the surfe re of eh tringulr prism. 15 m 15 m 9 m 15 m 50 m 4 m 6 m 5 m 5 mm 13 mm 24 m 15 mm 12 mm 3 m 10 m 3 m 10 mm 4 m 10 mm 2 m 10 mm 8 mm 6 m 21 mm 12 mm 9 m e 25 m f 8 m 7 m 24 m 25 m 40 m 7 m 8 m 5 m 11 m 18 m
43 504 CHAPTER 9: LENGTH, AREA AND VOLUME U N DE R S TA N DING A N D F LU E NC Y 7 Clulte the re of metl sheeting neee for this shipping ontiner. 3m 10 m 4m E 8 Clulte the surfe re of eh of these. M PL 29 m m m 32 m m SA 6 m m m 5.2 m 6 m 0.3 m 9 This toy ox is to e overe with oloure ontt pper. Wht re of pper will e neee to over oth the insie n outsie surfes of the ox? 09_NOL_MM7_TXT_SI.in m 0.5 m 15/07/14 11:51 AM
44 9G SURFACE AREA 505 PROBLEM SOLVING AND REASONING CHALLENGE 10 A ue hs surfe re of 150 m 2. Wht is the sie length of the ue? 11 A sulpture shpe s ue hs surfe re of 24 m 2. Wht re the imensions of the sulpture? 12 The outsie wlls n the roof of the og kennel re to e pinte. Clulte the re of the kennel to e pinte (in m 2 ). If smll n of pint overs 8 m 2, how mny ns of pint will e neee? 1.6 m 13 Another sulpture is onstrute in the shpe of retngulr ox with smller ox ple on top. The imensions of the oxes re shown in the igrm. To finish the sulpture, the rtist must work out the surfe re to e pinte n hene how muh pint is neee. Clulte the surfe re of the sulpture. (Ignore the surfe sitting on the groun s the viewing puli will not see this.) 1 m 2.6 m 70 m 80 m 1.8 m 1.4 m 40 m 1.2 m 0.8 m 2.2 m 14 These ojets re me of 1m loks. Work out their surfe re (inluing the se). 1 m 1.5 m Reflet Explin how re n still relte to 3D ojets s well s 2D shpes.
45 506 CHAPTER 9: LENGTH, AREA AND VOLUME 9H Volume n pity Strt thinking! Besies surfe re, other mesurements tht relte to 3D ojets re volume n pity. The volume of 3D ojet is the mount of spe it oupies. Common units use for volume re ui millimetres (mm 3 ), ui entimetres (m 3 ) n ui metres (m 3 ). Cui units re use sine the stnr units of volume re se on ues. 1 m A ue with sie length of 1 m hs volume of 1 m 3. 1 m 1 Desrie the imensions of ue tht woul represent: 1 m ui entimetre ui metre ui millimetre. 2 This retngulr prism is me up of 40 ues (two lyers of 20 ues). As eh ue hs volume of 1 m 3, the volume of the prism is 40 m 3. How mny ues re in one lyer? How mny lyers re there? Use your nswers to prts n to work out the volume of the prism. 5 m Wht is the length, with n height of the prism? e We n use shortut to lulte the volume. If the se of the prism hs n re of length with, show how you n use the shortut of multiplying the se re y the height to fin the volume of this prism. KEY IDEAS Volume is the mount of spe tht 3D ojet oupies. Common units of volume re ui millimetres (mm 3 ), ui entimetres (m 3 ) n ui metres (m 3 ). Volume of ny prism Rule: volume = se re height Formul: V = A h (or V = Ah) Volume of retngulr prism Rule: volume = se re height = length with height Formul: V = l w h (or V = lwh) The se n height of prism must e perpeniulr (t right ngles) to eh other. The mesurements for l, w n h must e in the sme unit efore sustituting into the formul. Cpity is the mount of flui tht ontiner n hol. Common units of pity re millilitres (ml), litres (L), kilolitres (kl) n meglitres (ML). l A 4 m w 2 m h
46 9H VOLUME AND CAPACITY 507 EXERCISE 9H Volume n pity EXAMPLE 9H1 Fining volume y ounting ues UNDERSTANDING AND FLUENCY Fin the volume of this retngulr prism. THINK 1 Fin the volume of eh retngulr prism. 5 m 4 m 2 m 3 m 3 m 3 m 2 m 4 m 4 m WRITE 1 Work out the numer of ues in one lyer. 4 3 = 12 ues in one lyer 2 Multiply y the numer of lyers to lulte the totl numer of ues in the prism. 3 Write the volume of the prism. Eh ue hs sie length of 1 m, so its volume is 1 m = 24 ues in prism volume = 24 m 3 3 m 7 mm 3 m 4 m 4 mm 2 mm e f 3 mm 3 m 6 mm 4 mm 5 m 3 m
47 508 CHAPTER 9: LENGTH, AREA AND VOLUME EXAMPLE 9H2 Clulting volume of retngulr prism using formul Clulte the volume of this retngulr prism. 10 mm 12 mm 7.5 mm UNDERSTANDING AND FLUENCY THINK 1 Ientify the 3D ojet (retngulr prism) n write the pproprite formul for volume. 2 Ientify the mesurements for l, w n h. Chek tht they re ll in the sme unit (yes). 2 Clulte the volume of eh retngulr prism. e 8 m 10 mm 2.4 m 7 m 4 mm 5 m 8 mm 1.5 m 0.8 m f WRITE V = l w h l = 12 mm, w = 7.5 mm, h = 10 mm 3 Sustitute the vlues for l, w n h into the formul. V = 12 mm 7.5 mm 10 mm 4 Clulte the result n inlue the pproprite unit. = 900 mm 3 6 m 3 m 3 m 3 m 9.1 mm 5 m 10.5 m 14.3 mm 7.8 mm 3 Clulte the volume of eh ue. 2 m 7 mm 2 m 2 m 7 mm 7 mm
48 9 H V O L UME AN D CAPACIT Y U N DE R S TA N DING A N D F LU E NC Y m 10 m 10 m 2.5 m 10 m 2.5 m 4 Clulte the volume of eh ojet. 30 m m m 50 mm E m m 60 mm M PL 30 mm 20 m 6 m 27 m m 6 5 The ojet t right hs its se re n height shown. SA The se shown is not the ottom surfe of the ojet n the height is not vertil mesurement. Does this mtter when lulting the volume? Explin. se re = 6 m2 Use the rule for the volume of ny prism to lulte the volume of this prism. height = 4 m 6 Clulte the volume of these prisms. The se re for eh prism is shown. se re = 4 mm2 se re = 8 m2 5 mm 10 m 7 m 12 mm 2 se re = 15 m 09_NOL_MM7_TXT_SI.in 509 se re = 40 mm 2 15/07/14 11:51 AM
49 510 CHAPTER 9: LENGTH, AREA AND VOLUME UNDERSTANDING AND FLUENCY 7 Clulte the volume of eh tringulr prism. (Hint: wht shpe is the se of eh prism?) 5 mm 12 mm 10 mm 4 m 7 m 6 m e f 5 mm 9 m 24 m 50 m 12 mm 15 mm 10 m 8 With prtner, list two exmples where mounts of liqui oul e mesure in the following pity units. ml L kl ML 9 Use the onversion igrm to omplete these sttements. 8 L = ml kl = L 70 ml = L ML kl L ml 6.4 ML = kl e 2500 L = kl f 3 kl = ml 10 The units of volume n pity re relte. A ontiner with n insie volume of 1 m 3 hols 1 ml of liqui. This mens tht ontiner with n insie volume of 1000 m 3 hols 1 L n ontiner with n insie volume of 1 m 3 hols 1 kl. Copy n omplete the tle y working out the missing vlue for eh ontiner. 2 m 5 m Insie volume of ontiner 2 m Cpity of ontiner 11 Soil hs een ug from gren to form retngulr hole of length 5 m, with 2 m n epth 50 m. How mny ui metres of soil hve een remove? 9 m 6 m 14 m 3 ml m 3 3 L 5.8 m 3 kl 750 m 3 L e m kl f m ML 12 A elivery of 3 m 3 of onrete is to e spre to form rivewy tht is 7.5 m long n 3.2 m wie. How thik will the lyer of onrete e?
50 9H VOLUME AND CAPACITY 511 PROBLEM SOLVING AND REASONING CHALLENGE 13 A sugr ue hs the imensions 1 m y 1 m y 1 m. Explin wht 1 m y 1 m y 1 m mens. Wht is the volume of one sugr ue? How mny sugr ues woul fit into ox with imensions 12 m y 10 m y 7 m? Desrie the imensions of ox tht oul extly fit 1000 sugr ues in it. Is there more thn one nswer? Explin. e Give five possile sets of imensions for ox tht oul extly fit 240 sugr ues in it. f How mny sets of imensions re possile for the smllest ox tht oul fit 20 sugr ues? List them. 17 m 14 Consier this ontiner of juie. Clulte the volume of the ontiner. Use your nswer to prt to lulte the pity of the ontiner. Is the pity of juie shown on the lelling of the ontiner the sme s your lulte vlue? If not, give resons why this might e. 10 m 6 m 15 Fin how mny litres of wter this retngulr sink will hol if fille to the top. 16 List the most suitle volume or pity unit for mesuring these mounts. the spe tken up y trin rrige the liqui in te up the hoolte in hoolte r the wter tht th n hol 1 Litre 17 Provie n estimte of eh mount esrie in question m 50 m 30 m 18 The fuel tnk of r hs pity of 50 L. How ig is the tnk? Suggest some possile imensions if the tnk is in the shpe of retngulr prism. 19 List two exmples of ifferent ojets tht hve volume lose to the following mounts. Explin your hoies. 10 m 3 10 m 3 Reflet 10 mm 3 Wht is the ifferene etween volume n pity?
51 512 CHAPTER 9: LENGTH, AREA AND VOLUME CHAPTER REVIEW SUMMARISE Crete summry of this hpter using the key terms elow. You my like to write prgrph, rete onept mp or use tehnology to present your work. twoimensionl (2D) shpe threeimensionl (3D) ojet units of mesurement length with MULTIPLECHOICE 9A 9B 9C 9D perimeter re retngle A = l w prllelogrm se height 1 Whih length mesurement is equivlent to 123 m? A 1230 mm B 12.3 m C km D m 2 The perimeter of this ojet is: A 6.5 m 4 m B 108 mm C 18 m D 29 mm 25 mm 3 A 1m gri hs een rwn over the shpe elow. The re of the shpe is losest to: A 8 m 2 B 14 m 2 C 20 m 2 D 22 m 2 4 The re of retngle with length 9 m n with 3 m is: A 6 m 2 B 12 m 2 C 24 m 2 D 27 m 2 9E 9F perpeniulr A = h rhomus tringle A = 1 2 h surfe re net 5 The re of this shpe is: A 14 m 2 B 24 m 2 C 49 m 2 D 144 m 2 prism volume pity V = A h V = l w h 6 The se length n height of this tringle re: A 8 m, 4.5 m B 8 m, 11 m 45 mm C 11 m, 4.5 m D 4.5 m, 8 m 8 m 9G 9H 7 The surfe re of this retngulr prism is: 4 m A 8 m 2 B 14 m 2 C 28 m 2 D 56 m 2 4 m 4 m 8 Whih pir of volume n pity mesurements re equivlent? A 40 ml n 4 m 3 B 1 m 3 n 1 L C 250 ml n 2.5 L D 500 m 3 n 0.5 L 11 m 1 m 6 m 2 m
52 9 CHAPTER REVIEW 513 SHORT ANSWER 9A 9A 9B 9B 9C 1 Convert these units. 34 m into mm 170 m into m 45 mm into m 8.7 km into m 2 Whih unit of mesurement woul e the est hoie to mesure eh of these? the length of penil the length of footll ovl the istne from your house to shool the length of fingernil 3 Estimte, then mesure, the perimeter of this shpe. 4 Clulte the perimeter of eh retngle. 5 m 0.8 m 25 m 30 m 5 A 1m gri hs een rwn over this shpe. Fin the pproximte re of the shpe. 9F 9G 9H 9.5 m 8 m 20 mm 8 Clulte the re of eh tringle. 7 m 25 m 24 m 45 mm 8 m 11 m 9 Clulte the surfe re of eh ojet. 75 m 110 m 35 m 20 mm 10 Clulte the volume of eh ojet shown in question 9. 9H 11 Clulte the volume of eh ojet. 400 m 3 m 5 m 9D 6 Clulte the re of eh retngle shown in question m 9E 7 Clulte the re of eh prllelogrm. 3 m 5 m 9H 220 mm 1.1 m 12 Fin the volume of retngulr prism if its pity is 0.75 L. Suggest set of imensions for the prism.
53 514 CHAPTER 9: LENGTH, AREA AND VOLUME NAPLANSTYLE PRACTICE 1 Kyle knows tht it is extly 798 m from his house to shool. Convert this istne to kilometres. km 2 Whih length mesurement is the lrgest? 6500 mm 5600 m 7.5 m km 3 Wht is the perimeter of this shpe? 25 mm 2 m 4 m 20 mm 2.5 m 53.5 mm 13 m 22 m mm 4 Wht is the perimeter of squre of length 8 mm? mm 5 A 1m gri hs een ple over this shpe. Wht is the pproximte re of the shpe? m 2 6 Wht is the re of squre of length 8 mm? mm 2 Questions 7 n 8 refer to this retngle. 60 mm 0.5 m 7 Wht is the re of the retngle? 30 mm 2 3 m 2 30 m mm 2 8 Wht is the perimeter of the retngle? 30 mm 13 m 30 m 121 mm 9 A retngle hs length of 30 mm n n re of 24 m 2. Wht is the with of the retngle in entimetres? m Questions refer to this shpe. 10 Wht is the nme of the shpe? retngle prllelogrm 2 m 35 mm 4.5 m tringle squre 11 Wht is the perimeter of the shpe? 65 mm 9 m 110 mm 13 m 12 Wht is the re of the shpe? 130 mm 2 7 m 2 9 m mm 2 Questions 13 n 14 refer to this shpe. 1.3 m 1.2 m 13 Wht is the perimeter of the shpe? 2.5 m 3 m 7.5 m 52.5 m 14 Wht is the re of the shpe? 30 m 2 60 m m m 2 15 All of the tringles in this shpe re ientil. Wht is the re of the entire shpe? 6 m 3 m 2 30 m 2 45 m 2 60 m 2 50 m 6 m
54 9 CHAPTER REVIEW The re of squre is 49 m 2. Wht is its perimeter? m 19 Wht is the volume of the vse? 50 m 3 62 m m m 3 17 Whih shpe hs the lrgest re? ANALYSIS retngle with length 70 mm n with 3 m tringle with se length 10 m n height 40 mm squre with length 5 m prllelogrm with se length 8 m n height 30 mm Questions refer to this ojet. Ell ought vse for her new house. 50 m 18 Wht is the surfe re of the outsie of the vse? 6 m 6 m 62 m m m m 2 For frien s irthy, you eie to uy portle DVD plyer. There re three within your prie rnge, with slightly ifferent sreen imensions. Option 1: height 8 m y with 16 m Option 2: height 9 m y with 15 m Option 3: height 110 mm y with 140 mm 20 Wht is the mount of wter tht this vse oul hol? 1.8 L 18 L 180 L 1800 L Questions 21 n 22 refer to new storge room of length 6 m, with 4 m n height 3 m. The storge room nees to e pinte. There is oor to the room with imensions 200 m y 150 m tht hs lrey een pinte. 21 Wht is the re of the wlls, floor n eiling tht woul nee to e pinte? m 2 22 Wht volume of goos oul e store in the room? m 3 You eie to selet the DVD plyer tht hs the wiest sreen imension. Wht isvntge oes this hve ompre to the other two options? If the DVD plyer is only 3 m thik, wht is the volume of the smllest ox tht you oul put it in? The shop ssistnt tells you tht ll three hve the sme sreen perimeter. Fin the perimeter of eh sreen. Is the shop ssistnt orret? A frien tells you tht if the sreens hve the sme perimeter, they must hve the sme re. Clulte the re of eh sreen n explin why this is not true. e The DVD plyer omes in ox tht mesures 100 mm y 200 mm y 50 mm. Wht is the volume of this ox? f You will nee to wrp the present. Clulte the surfe re of the ox so tht you know the minimum mount of pper you will nee. g Why might you nee it more wrpping pper thn the minimum? Think out how you wrp ojets.
55 516 CHAPTER 9: LENGTH, AREA AND VOLUME CONNECT Plnning swimming pool Your tsk To uil n ingroun swimming pool requires plnning n some reful mthemtil lultions. Wht mesurements re neee when plnning the uiling of kyr swimming pool? A retngulr setion of ln of length 12 m n with 9 m hs een lere for n ingroun swimming pool. This ln lso inlues spe for pving roun the pool n sfety fene. The floor of the pool will e horizontl. This mens tht the epth of the wter will e the sme ross the whole pool. The onrete wlls n floor of the pool re to e pinte lue. 12 m 12 m 12 m 9 m 9 m 9 m Possile plns for swimming pool (eril view) Your tsk is to esign your own swimming pool n lulte eh of these mesurements: the mount of pving require the length of sfety fening neee the mount of soil to e remove to form the hole for the pool the re of the wlls n floor of the pool to e pinte the mount of pint neee to pint the pool the mount of wter neee to fill the pool the mount of plsti mteril neee for the pool over ny other mesurements you think re neee. Another option for the esign of n ingroun swimming pool is to hve the epth of the pool vry from one en to the other, Sie view of pool s shown in this igrm. How woul this hnge ffet your mesurement lultions? Inlue ll neessry working to justify your nswer.
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