Chpter 11 Mesurement II Wht you will lern 11-1 The metri system 11-2 Perimeter of polygons 11-3 Perimeter of irles 11-4 Are of polygons 11-5 Are of irles nd ompounded shpes 11-6 Volume of prisms 11-7 Prtil mesurement Westp Stdium, Wellington Westp Stdium is the fous for interntionl sporting fixtures nd for mny outdoor musi nd ulturl presenttions in New Zelnd s pitl. It ws uilt on the foreshore on lnd previously used y NZ Rilwys. It is loted 680 m from the rilwy sttion. In n emergeny, evution n e hieved in 8 minutes. The light tower is 55 m high. The sports owl is 48000 m 2 nd the roof is 12 000 m 2. 7930 tonnes of snd nd 2800 tonnes of dringe grvel were used to mke the pith. 900 kilogrms of grss seed ws sown. For rugy, the height of the grss is 20 25 mm, ut for riket it is redued to 12 13 mm. 85 000 litres of wter re used for wtering eh dy. If ll is hit from the entre riket wiket to outside the ground, it trvels distne of 98 metres. During its onstrution, nd even now in its use, onsidertion of length, re, volume, mss, pity nd time re ll required. 384
New Zelnd Curriulum LEVEL 4 Geometry nd Mesurement Use pproprite sles, devies, nd metri units for length, re, volume nd pity, weight (mss), temperture, ngle, nd time Convert etween metri units, using whole numers nd ommonly used deimls Use side or edge lengths to find the perimeters nd res of retngles, prllelogrms, nd tringles nd the volumes of uoids LEVEL 5 Geometry nd Mesurement Selet nd use pproprite metri units for length, re, volume nd pity, weight (mss), temperture, ngle, nd time, with wreness tht mesurements re pproximte Convert etween metri units, using deimls Dedue nd use formule to find the perimeters nd res of polygons nd the volumes of prisms Find the perimeters nd res of irles nd omposite shpes nd the volumes of prisms, inluding ylinders 385
Do now 1 Clulte: 3.4 + 2.8 + 4.6 + 2.2 56 4.29 26.8 + 26.4 2 Clulte: 6 8 121 0.5 8.3 2.3 3 Clulte: 28.96 100 497.1 1000 8.31 10 4 Clulte: 4250 100 28.96 10 000 68.54 100 5 Complete: 1 km = m 5 km = m 0.8 km = m 6 Clulte the re y ounting the numer of squres. Prior knowledge metre entimetre length litre grm kilometre re volume millilitre kilogrm millimetre 7 Whih of the shpes in Question 6 hs the lrgest re? 8 Clulte the numer of oxes required to uild these shpes. 9 Evlute: 5 2 2 3 3 3 d 4 2 10 A mp of the South Islnd shows six regions. By ompring the size of eh region, rrnge them in order from smllest to lrgest. 386 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
11-1 The metri system In 1795 the first drft of the metri system ws mde legl in Frne. A metre ws defined s one ten-illionth of the distne from the North Pole to the equtor going through Frne. In 1969 New Zelnd first strted eduting the puli on the metri system, whih eme lw in 1976. Slly weighs g of fruit nd the sle displyed mss of 1320 g. She would like the mss of the fruit in kilogrms, ut Slly is unsure if the mss is 1 320 000 kg or 1.320 kg. How mny grms re in 1 kilogrm? Whih nswer do you think is orret? Disuss. Key ides The metri system is used to mesure mss, length nd volume. Length is mesured in millimetres, metres nd kilometres. 10 mm = 1 m 100 m = 1 m 1000 m = 1 km Mss is mesured in milligrms, grms, kilogrms nd tonnes. 1000 mg = 1 g 1000 g = 1 kg 1000 kg = 1 t Cpity is mesured in millilitres, litres, kilolitres nd meglitres. 1000 ml = 1 L 1000 l = 1 kl 1000 kl = 1 ML Conversion hrts Length mm Cpity ml 10 100 1000 m 10 100 1000 m km Mss mg 1000 1000 1000 1000 1000 1000 L kl 1000 1000 1000 ML g kg 1000 1000 1000 t Chpter 11 Mesurement II 387
EXAMPLE 1 Convert these mesurements. 7.32 m to m 0.032 t to kg 5600 L to kl Solution 732 m 32 kg 5.6 kl EXAMPLE 2 Explntion 1 m = 100 m. 7.32 m is etween 7 m nd 8 m so the nswer is going to e etween 700 m nd 800 m. We n use doule numer line: 0 0 7.32 m 100 200 300 400 500 600 700 800 m 1 2 3 4 5 6 7 8 This doule numer line shows tht 7.32 m is the sme s 732 m. We ould lso use prt of the length onversion hrt. This sys to multiply y 100. 7.32 100 = 732 1 t = 1000 kg, ut we hve less thn 1 t, so the nswer is going to e less thn 1000 kg. We ould use prt of the mss onversion hrt. This sys to multiply y 1000. 0.032 1000 = 32 7.32 m 1000 L = 1kL, 5600 L is etween 5000 nd 6000, so the nswer is going to e etween 5 kl nd 6 kl. We ould use doule numer line or prt of the pity onversion hrt. This sys to divide y 1000. 5600 1000 = 5.6 m kg L m 100 100 1000 1000 1000 1000 Eli is mking 6 dozen muffins, eh dozen of muffins requires 200 ml of milk. How mny litres of milk does Eli need? Solution 6 200 = 1200 ml 1200 ml = 1.2 L Eli requires 1.2 L of milk. Explntion First we need to find out how muh milk is required in millilitres. Eli requires 6 lots of 200 ml 6 200 = 1200 ml 1200 ml is the sme s 1.2 L 388 m t kl Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exerise 11 Exmple 1 Exmple 1 Exmple 1 1 Mth these ojets to the pproprite mesurement unit. length of footll field mss of vitmin tlet pity of uket d distne from Auklnd to Wellington e length of n nt f pity of milk tnker g mss of n dult elephnt i mg ii m iii L iv t v km vi kl vii mm 2 Copy nd omplete these digrms. 100 mm 1000 1000 L 3 Convert these to metres. A door tht is 1920 mm high The distne from New Plymouth to Okur is 13.8 km One-yer-old Leonie is 60 m tll 4 Complete these onversions. seven metres = millimetres three metres = entimetres 0.75 kilometres = metres d 1.08 m = m e 0.065 km = m f 56 000 mm = m = km g 3.1 km = m = m 5 Convert to litres. A petrol tnker with pity of 6 kl A ottle of wter tht holds 750 ml A owl tht hs pity is 2500 ml km mg 1000 g Chpter 11 Mesurement II 389
Exmple 1 Exmple 1 Exmple 1 Exmple 2 Exmple 2 Exmple 2 Exmple 2 Exmple 2 6 Complete these onversions. three litres = millilitres five litres = millilitres four thousnds millilitres = litres d seven kilolitres = litres e 5.7 litres = millilitres f 840 meglitres = kilolitres g 100 ml = L h 5000 L = ml i 0.98 kl = ML j 0.06 L = ml k 58.4 ml = L l L = ml 6 1 2 7 Convert these to grms. A rhinoeros eetle with mss of 83 900 mg Mss of y 3.98 kg Mss of n elephnt 0.9 t 8 Convert these mesurements. five kilogrms = grms three hundred milligrms = grms sixteen kilogrms = grms d 874 mg = g e 4.5 kg = g f 7500 g = kg g 340 000 g = kg = t h 5 kg = g = mg 9 Write these mesurements in order from smllest to lrgest. 50 m, 60 m, 4500 mm, 6 km, 500 m 0.25 kg, 500 g, 0.025 t, 250 mg, 5000 mg 19 L, 0.19 kl, 7000 ml, 0.0091 ML, 7.5 L d 85 mg, 8.5 kg, 0.72 t, 7200 g, 58 000 mg e 45 mm, 540 m, 4.5 m, 0.054 km, 5.4 m 10 Five friends shre 350 g hoolte r. How muh does eh person reeive? 11 At Rit s irthdy prty 5.4 L of flvoured drink is in the punh owl. If eh glss n hold 150 ml, how mny glsses of drink re provided? 12 Bords used to form grden edging re stood vertilly nd pled side y side. The grden is 9.25 m long nd 74 ords re required. How wide is eh ord? 13 The dimeter of 50 oin is 19 mm. The oins re pled side y side in oin tril on footpth. If the tril is 9.5 m long, wht is the vlue of the oins on the footpth? 14 Georgi n purhse 120 g pket of heese for $2.50 or 300 g pket for $5.89. How muh does eh pket of heese ost per 100 g? Whih pket of heese is the est vlue for money? 390 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exmple 2 Exmple 2 15 A truk is le rry mximum lod of 15 tonnes. How mny rtes, eh hving mss of 12 kg, n e rried y the truk? Cn the truk rry lod of 80 television sets (17 kg eh), 60 DVD reorders (7.5 kg eh) nd 150 musi systems (11 kg eh)? If the truk is rrying etween 1 nd 1 1 tonnes of televisions, how mny televisions n there e? 2 16 Clulte the totl mount of milk in these jugs. (Give your nswer in litres.) ENRICHMENT: Mesuring pge 17 It is diffiult to mesure the thikness of single pge, ut we ould use mny different methods to help us. Desrie how you ould otin mesurement of one pge of your textook. Mesure in mm the thikness of one pge of: i your textook ii your exerise ook iii newspper iv your fvourite mgzine. Convert eh mesurement into: i entimetres ii metres. d Wht is the differene etween the thinnest nd thikest pge? Chpter 11 Mesurement II 391
11-2 Perimeter of polygons The perimeter is the distne round the outside edge of two-dimensionl ojet. To lulte the perimeter, Peter suggests this formul: Nellie suggests this formul: Perimeter = 6 + 1 + 4 + 4 + 2 + 5 Perimeter = 2 16 + 52 Investigte different wys the perimeter of this shpe ould e lulted. Key ides To find the perimeter of ny polygon you dd the lengths of ll the sides. The mesurements must e expressed in the sme unit efore they n e dded. The units of perimeter re millimetres, entimetres, metres nd kilometres. It is importnt to write the unit of the nswer. 8 n men mny different things, ut 8 m gives more informtion. EXAMPLE 3 Clulte the perimeter of these polygons. 2 km 3 km 5 km Solution 4 km 2 km Perimeter = 2 + 3 + 2 + 4 + 5 = 16 km Perimeter = 6 + 1 + 4 + 4 + 2 + 5 = 22 m 50 mm Explntion 2 m 6 m 10 mm First hek tht ll the sides re in the sme unit. Now we n dd them together. Perimeter = 2 + 3 + 2 + 4 + 5 Not ll these sides re in the sme unit. Rememer: 10 mm = 1 m 50 mm = 5 m There re two sides tht re unlelled. Now tht we hve ll the informtion we n lulte the perimeter. 5 m Perimeter = 6 + 1 + 4 + 4 + 2 + 5 = 22 m 50 mm 2 m 6 m 2 m 6 m 5 1 = 4 m 1 m 6 2 = 4 m Is there nother wy tht we ould solve this perimeter prolem? 392 10 mm Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exmple 3 EXAMPLE 4 Clulte the unknown length. 5 m 7 m 4 m x m perimeter = 19 m Solution Explntion x = 3 m 4 + 5 + 7 + x = 19 16 + x = 19 1 + 1.5 + 1.5 + x + 1.5 + 1.5 + 1 + 1.5 + 1.5 + x + 1.5 + 1.5 = 20 14 + 2x = 20 2x = 6 x = 3 m Exerise 11 1 Clulte the perimeter of eh shpe. 38 m 20 m 1 m x must equl 3 m, sine 16 + 3 = 19. 1.5 m x m perimeter = 20 m It is importnt to rememer the unit of the nswer. On this imge there re some dshes on the sides of the polygon. All the lines with one dsh re the 1.5 m, ll the lines with two dshes re 1 m nd ll the lines with three dshes re unknown. Perimeter: 1 + 1.5 + 1.5 + x + 1.5 + 1.5 + 1 + 1.5 + 1.5 + x + 1.5 + 1.5 = 20 14 + 2x = 20 x m 1.5 m 1.5 m 1 m 1 m 1.5 m 1.5 m The two unknown sides must equl 6, so one unknown side must e 3. 24 m 70 m 68 m 20 m 28 m 4 m x m Chpter 11 Mesurement II 393
Exmple 4 e 620 m 2.6 m 7.4 m 1.4 m 5 m 5.2 m 5 m 8 m 480 m 4.8 m 5.2 m d f 55 mm 7.9 m 2 Drw retngle with perimeter of 40 m. Drw different retngle with the perimeter of 40 m. Drw nother different retngle with the perimeter of 40 m. 2 m 8.9 m 8.4 m 65 mm 10.3 m 3 Drw equilterl tringle with the perimeter of 24 m. Drw n isoseles tringle with perimeter of 24 m. Wht is the sme nd wht is different out the two tringles tht you drew in questions nd? 4 The length of retngulr pddok is 25 m nd its width is 20 m. Wht is the totl perimeter of the pddok? A fmily room in disply home is squre in shpe with side length of 4.1 m. Wht is the perimeter of the fmily room? A pond is in the shpe of rhomus of side length 2.4 m. Wht is the pond s perimeter? 5 Clulte the missing vlue x. 7 m 2 m x perimeter = 16 m 3 m 40 m 18 m 32 m x perimeter = 135 m 20 m 10 mm 394 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
e 18 m x m 6 m perimeter = 52 m 17 m 19 m x m perimeter = 84 m 6 Brend wnts to uild fene round her retngulr grden of length 18 m nd width 24 m. She pys $28 for eh fene post. If the posts re 2 m prt, how muh will it ost for fene posts? 7 An deides to wrp eh gift tht she sells with red rion. She hs estimted tht eh present requires 80 m of rion. If An hs 12 m of rion, how mny gifts n she wrp? ENRICHMENT: Polygon frmes d 5.5 m perimeter = 17 m 8 Dimond Frming displys this digrm t the front desk. It gives the prie of eh frme. i iv ii $ 400 $ 400 = 100 m 2 $ 500 iii How hve Dimond Frming deided to prie their piture frmes? One of the frmes is inorretly pried. Whih frme is it nd wht is the orret prie? Leon would like to mke speil request for frme with dimension 90 m y 50 m. How muh does Leon s frme ost? v $ 560 $ 480 x m Chpter 11 Mesurement II 395
11-3 Perimeter of irles Leon nd Poppy were mesuring the irumferene nd dimeter of ylinders. They put the informtion into tle. Ojet Cirumferene (C ) Dimeter (D) 1. n of drink 21 m 6.6 m 2. n of fish 21.2 m 6.7 m 3. ke tin 62.7 m 20 m 4. pint tin 56.3 m 18 m Wht do you notie out the results in the C D olumn? Copy nd omplete this sttement: The irumferene of eh ojet is pproximtely times the dimeter. Key ides Prts of irle. rdius dimeter irumferene Cirumferene = π dimeter C = π d Cirumferene = π two lots of rdius C = 2 π r For most lultions n pproximtion of 3.14 n e used for π. Cirumferene dimeter (C D) 396 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
EXAMPLE 5 Clulte the irumferene of these irles. 5 m 8 m Solution C = π 8 C = 25.13 m C = 2 π 5 C = 31.42 m EXAMPLE 6 Explntion These irles re not drwn to sle so we must use the formul C = π d to lulte the irumferene. Dimeter is 8 m. A dimeter n fit into the irumferene pproximtely 3.14 (or π) times. C = π 8 or C = 3.14 8 Two rdii form the dimeter. So insted of π d we must multiply the rdius y 2 nd then multiply y π. C = 2 π 5 or C = 2 3.14 5 Wht is the dimeter of the lrgest irle tht n e mde from: 44 m of wire? 80 m of rion? Solution d = 44 π d = 14 m d = 80 π d = 25.46 m Explntion Approximtely 3.14 dimeters fit into the irumferene. So we must find how mny dimeters re in 44 m. d = C π d = 44 π d = C π d = 80 π d = C 3.14 d = 44 3.14 d = C 3.14 d = 80 3.14 Note: ll nswers in these exmples hve een rounded to 2 deiml ples. (Two digits fter the deiml point). Chpter 11 Mesurement II 397
Exerise 11 Exmple 5 1 Copy nd omplete the tle. i Using piee of string, mesure the irumferene of eh irle. ii Using ruler, mesure the dimeter of eh irle. Cirle Cirumferene (C ) Dimeter (D) d e f e f d Cirumferene dimeter (C D) 2 Mth the irle to the orret eqution to lulte the irumferene. e 6 m 6 m 9 m d f 2.5 m i C = 9 π ii C = 4 2 π iii C = 6 π iv C = 2 6 π v C = 2 9 π vi C = 2.5 π 398 2 m 9 m Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exmple 5 3 Clulte the irumferene of eh irle (round your nswer to 2 deiml ples). f 1.5 m g h PL E e 14 m 5 m 7 m 2 m d 0.9 m 3.8 m 2.4 m Exmple 6 4 How muh wire (to the nerest metre) is needed to form irle of dimeter 30 metres? Exmple 6 5 A irulr plyground hving dimeter of 7.64 m is to e fened. How mny metres of fening is needed? Give your nswer orret to the nerest metre. If fening osts $15 metre, wht is the ost of fening the plyground? Posts re needed out every 2 metres round the fene. How mny posts re needed? 6 The minute hnd on wth is 1.6 m long. How fr does its tip trvel (to the nerest m) in: 1 hour? 1 dy? 7 A iyle wheel hs dimeter of 63 m. Clulte the irumferene of the wheel to the nerest entimetre. How fr, to the nerest metre, will the iyle trvel if the wheel turns: i 50 times? ii 1000 times? iii 5600 times? SA M Exmple 6 Exmple 6 8 The irumferene of metl rod is 6 m. Will the rod go through the hole in the lok shown? Explin your nswer. 1.6 m 399 Chpter 11 Mesurement II 978110762840311_p384-427.indd 399 09/08/13 11:35 PM
Exmple 6 9 If Tim wlks 1 km in 10 min, how long does it tke him to wlk round irulr pth if the irle hs dimeter of 2.4 km? 10 The irumferene of this irle is 28 m. Wht is its dimeter? (Round to whole numer.) Wht is its rdius? 11 Leonie is designing grden ed, ut she nnot quite mke up her mind out its shpe. She hs nrrowed the hoie down to the designs shown elow. 2 m 5 m 3 m 3 m i Find the totl distne round the edges of eh grden ed, orret to two deiml ples. ii If Leonie is going to edge eh grden ed with terrott order t ost of $24 per metre, wht is the ost of edging eh ed? iii Whih shpe would you reommend? Why? d 2 m 4 m 4 m C = 28 m 400 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
d e Wht shpe ws your pth? How fr did you wlk? Use the reltionship you found etween the irumferene nd the dimeter, in the first tivity, to help you. Chek your nswer y mesuring your pth. Think out how you n do this. Repet the sme tivity using the other piees of string. Ask yourself the sme questions eh time. Reord ll your results in tle similr to the one elow. Length of string (rdius of the irle) Dimeter of the irle 13 Copy nd omplete this sttement: Estimte of irumferene The irumferene of eh ojet is pproximtely times the rdius. ENRICHMENT: Chnging the dimeter Mesurement of the irumferene 12 You will need four piees of string (eh different length) nd ruler, tpe mesure or trundle wheel. Mesure the length of eh piee of string. Your prtner needs to hold the end of one of the piees of string nd stnd still. You need to hold the other end. Keep the string tut nd level nd strt wlking round. Your prtner stys in the one spot, ut needs to turn on the spot to keep fing you. Stop when you hve returned to your strting point. Chpter 11 Mesurement II 401
11-4 Are of polygons My is trying to lulte the re of prllelogrm. She knew to lulte the re of retngle you multiply se y height. She thought the se of the prllelogrm is 5 m, nd height 3.6 m. This gve n re of 18 m 2. Just to hek her nswer she ounted the numer of squres, whih gve her n re of 15 m 2. Whih one is the orret nswer? Explin your nswer. Did My use the orret mesurements? Explin your nswer. Key ides Are is the spe tht two-dimensionl ojet tkes up. Units for re: millimetres squred (mm 2 ), entimetres squred (m 2 ), metres squred (m 2 ), kilometres squred (km 2 ) Lnd re is mesured in hetres (h). 10000 m 2 = 1 h. The re of ommonly used polygons re: Length (l ) Are = length width Are = l w Width (w ) Clulte the re of these shpes. EXAMPLE 7 8 m 5 m Bse () Height (h) Are = se height Are = h 6 m 4 m 3 m Height (h) 5 m Bse () Are = 1 se height 2 Are = 1 2 h 402 100 mm 50 mm 3.6 m Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Solution Explntion Length is 8 m nd the width is 5 m. 8 m Are = 8 5 Are = 40 m 2 5 m In eh row there is 8 m 2. There re 5 rows of 8 m 2, so there is 5 lots of 8 m 2. To lulte the re we multiply 8 y 5. Are = length width Are = l w Units of mesure re m 2. A = 1 2 6 4 1 Are = 2 h A = 12 m 2 Bse = 6 m, height = 4 m. A = 100 50 A = 5000 mm 2 EXAMPLE 8 Units of mesure re m 2. Note: the height must e t right ngles to the se. Are = h Bse = 100 mm, height = 50 mm. Units of mesure re mm 2. Note: the height must e t right ngles to the se. Theo is going to pint retngulr rik wll. The dimension of the wll is 8 m y 2 m. One litre of pint will over 2.5 m 2. How muh pint will Theo need to pint the rik wll? Solution A = 8 2 A = 16 m 2 8 m Amount of pint = 16 2.5 = 6.4 Theo needs 6.4 litres of pint. 2 m Explntion It is lwys good to drw digrm. The digrm should inlude ny dimensions tht re given in the question. First we need to lulte the re of the retngle. A = h Units of mesure re m 2 1 L of pint overs 2.5 m 2, we need to lulte how mny 2.5 re in 16. Chpter 11 Mesurement II 403
Exerise 11d Exmple 7 1 Choose the most suitle unit (mm 2, m 2, m 2, km 2, h) for mesuring the re of eh of these items. the top of your desk 10-ent oin diry frm d Wellington e your shool s lnd f fingernil g postge stmp h silver fern frond i this pge j your hnd 2 Estimte the re of eh shpe y ounting the numer of squre units. d e f 3 For eh of these tringles: i urtely mesure the dimensions ii lulte the re. 404 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exmple 7 Exmple 7 4 For eh of these retngles nd prllelogrms: i urtely mesure the dimensions ii lulte the re. e 5 Clulte the re of these retngles nd prllelogrms. These shpes re not drwn to sle. 6 m 8 m 10 m d 7 m Chpter 11 Mesurement II 405
Exmple 7 e 110 mm 6.7 m 4.6 m 50 mm d 32 mm 6 Clulte the re of these tringles. These shpes re not drwn to sle. 64 mm 13 m 9 m 82 mm 7 Copy nd omplete the following tle. Length Width Are of retngle 4 m 3 m 5 m 10 m 2 6 m 18 m 2 7 km 49 km 2 8 m 24 m 2 d 4 m 3.2 m 3 m 5 m 48 mm 8 Drw retngle with n re of 36 m 2. Drw different retngle with n re of 36 m 2. Drw tringle with n re of 36 m 2. d Drw different tringle with n re of 36 m 2. e Wht is the sme nd different out the retngles nd tringles tht you drew? 2.5 m 406 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exmple 8 Exmple 8 Exmple 8 9 How mny squre pvers, if eh of side length 20 m, re needed to over n re of 20 m 2? 10 One pket of lwn seed overs 4 m 2. How mny pkets re needed to plnt lwn on retngulr re tht mesures 9 m 8 m? If the re of lwn ws douled, how mny pkets of lwn seed do you need? If the re of lwn ws hlved, how mny pkets of lwn seed do you need? 11 A netll ourt hs dimensions 30.5 m y 15.5 m, nd sketll ourt hs length of 28 m nd width of 15 m. Find the re of eh ourt. Whih ourt hs the lrger re? A new stdium is eing uilt. i Cn you fit two netll nd two sketll ourts in the new stdium ourt re if the length is 62 m nd the width is 34 m? ii Drw digrm to show how this ould e done. 12 Mtij is mking photo frmes. The photos re 10 m y 8 m. The frmes re 2 m wide. Wht is the re of the photo? Wht is the re of the whole frme? Wht is the re of the frme only? ENRICHMENT: Are of skin on your hnd 13 In generl, the re of 100 hnds (fingers together) pproximtely overs your ody. Medil stff use the re of the hnd to estimte the re of skin urned on ptients. Tre the outline of your hnd onto grid pper. Find the re of your hnd. Use your hnd re to estimte: i the re of your fe ii the re of your rms nd legs iii the re of ll your ody. Chpter 11 Mesurement II 407
11-5 Are of irles nd ompound shpes Cinnmon divided irle into 20 equl setors nd rrnged them like this. Wht shpe do the setors mke? Ben noties tht the shpe is very similr to retngle. He lso noties the height of the retngle is the rdius of the irle. Ben ut the shpe into smll squres with the width the sme s the rdius. How mny rdius squres does the irle fit into? Investigte. Key ides Are of irle = π r 2 To lulte the re of ompounded shpe, divide the shpe into retngles, tringles or semi-irles. = + To lulte the re of the shded region, sutrt the re of the shpes. EXAMPLE 9 = Clulte the re of these irles. 5 m 40 mm 408 = + Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Solution Explntion A = π 5 2 A = π 25 A = 78.54 m 2 A = π 20 2 A = π 400 A = 1256.64 mm 2 EXAMPLE 10 To lulte the re of irle we use the formul A = π r 2. π is n irrtionl numer tht goes on forever. π = 3.1415926535897932384 A sientifi lultor hs π utton, or we ould just use the pproximtion of 3.14. Rdius = 5 m 3.14 25 The units re is m 2 Dimeter is 40 mm so the rdius is hlf the dimeter. r = 20 mm Clulte the re of these shpes. 12 m Solution 8 m A 1 = 12 4 A 1 = 48 m 2 A 2 = 8 4 A 2 = 32 m 2 Totl re: Are = 48 + 32 Are = 80 m 2 Explntion 25 mm 30 mm 40 mm 25 mm 50 mm The dshes on the shpe tell us whih sides re the sme lengths. One method of lulting the re is y dding the re of two retngles. 12 m Are = 12 4 = 48 m 2 Are = 8 4 = 32 m 2 8 m 4 m 12 8 = 4 m To lulte the re of the whole shpe, dd the two res together. Is this the only method of lulting the re? Chpter 11 Mesurement II 409
Solution Are = 2 (25 30) + 90 20 Are = 1500 + 1800 Are = 3300 mm 2 Explntion This shpe n e divided into three retngles. 25 mm 30 mm 40 mm 25 mm 50 mm Two of the retngles re the sme. Are ould e lulted y: Are = + 2 30 mm 25 mm 90 mm Are = 2 (25 30) + 90 20 Alwys rememer the units. Clulte the shded region of these shpes. EXAMPLE 11 3 m 2 m Solution Are = 3 3 π 1 2 Are = 9 3.142 Are = 5.858 m 2 3 m 5 m 10 m 8 m 20 mm 7 m Explntion To lulte the re of the shded region we must first lulte the re of eh shpe nd then sutrt the res. Are = Are = h π r 2 Are = 10 7 8 5 Are = 70 40 Are = 30 m 2 Are = Rememer units of mesure. 410 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exerise 11e Exmple 10 Exmple 9 1 For eh shpe: i urtely mesure the dimension of the shpes ii lulte the re of the totl shpe. 2 Eh irle is drwn to sle. For eh irle: i estimte the re of eh irle ii lulte the re of the irle using the formul A = π r 2. d d 1 m Chpter 11 Mesurement II 411
Exmple 10 Exmple 11 3 Clulte the re of eh omposite shpe. These shpes re not drwn to sle. 2 m e 5 m 7 m 3 m 6 m 7 m 12 m 5 m 5 m 10 m 16 m 2 m 18 m 4 Clulte the shded region. 2 m 3 m 1 m 6 mm d 5 m 4 m 9 m d 45 mm 6 m 85 mm 2 mm 5 mm 6 mm 6 m 3 m 2 m 1 m 7 m 10 m 9 mm 412 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exmple 9 5 Clulte the re of these shpes. 8 m 70 mm 6 A rug 2.8 m y 3.2 m is pled in squre living room with the dimensions of 4 m y 4 m. Drw ird s-eye view of the living room floor. Write the dimensions on the digrm. Clulte the re of the floor tht is unovered. 7 The outer edge of piture frme mesures 48 m y 32 m. The photogrph inside the frme is 28 m y 24 m. Drw digrm of the photo frme. Write the dimensions on the digrm. Clulte the re of the order round the photo frme. 8 A irulr grden ed with dimeter of 3 m is surrounded y pth with width of 1 m. Clulte the re of the pth. 9 A retngulr grden ed 5 m y 8 m is surrounded on three sides y 1 m pth. Drw digrm of the grden. Write the dimensions on the digrm. Clulte the re of the pth 10 m Chpter 11 Mesurement II 413
10 Georgi hs ought ity prtment onsisting of five min rooms s shown in the house pln. d e 3.6 m 3.8 m 1.9 m Bedroom 2 Bedroom 1 Bthroom 2.5 m 4.3 m Lounge/Dining Kithen 4.2 m 4.3 m Clulte the re of eh min room. Clulte the re of the whole prtment. If slte floor tiles ost $75 per squre metre, lulte the ost of tiling the kithen nd lounge/ dining room. If wool rpet osts $120 per squre metre, lulte the ost of rpet for the edrooms. Whih osts more: tiling the living res or rpeting the edrooms? ENRICHMENT: Squres in squres 11 All the setions in this shpe re squres. The re of D is 81 m 2. The re of E is 100 m 2. Clulte the re of eh squre. Is the totl shpe squre? Investigte. B A H G C I F E D 4 m 3 m 414 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
11-6 Volume of prisms Is nd Alie re hiring truk to deliver 12 offee tles. The offee tles re in oxes tht re 1 m y 0.8 m y 1.2 m. They re given three different types of vehiles to selet from. Truk A: 3 m y 2 m y 2 m Truk B: 2 m y 1.5 m y 4 m Vn: 3.2 m y 1.6 m y 1.4 m (The dimensions re length y width y height.) Whih vehiles n hold volume of 12 m 3? Investigte whih vehile Is nd Alie ould use tht ould fit in ll twelve offee tles. (You my wish to drw digrm or use loks to represent the offee tles.) Key ides Volume is mesure of the mount of spe three-dimensionl shpe oupies. Cpity is the mount of something n ojet n hold. The most ommon mesures of volume re: ui metres (m 3 ) ui entimetres (m 3 ) ui millimetres (mm 3 ). To lulte the volume of ny three-dimensionl shpe V = re of se height Volume of two ommon three-dimensionl shpes Cuoid Tringulr prism height height width width length length Volume = l w h Volume = 1 2 l h w Are of se Height An ojet with the dimension 1 m y 1m y 1m hs volume of 1 m 3. Chpter 11 Mesurement II 415
EXAMPLE 12 Clulte the volume of these shpes. 9 m Solution A = 24 m 2 V = 24 8 V = 192 m 3 V = 12 10 9 V = 1080 m 3 EXAMPLE 13 8 m Clulte the volume of these shpes. 6 m Solution V = 6 6 6 V = 216 m 3 6 m 6 m Are of tringle = 1 35 40 2 = 700 mm 2 Volume = 700 70 = 49 000 mm 3 12 m Explntion 10 m The volume formul for three dimensionl prism is: V = re of se height. Are of se = 24 m 2 Height = 8 m Rememer the units of mesure. The volume formul for uoid is: Volume = l w h. Length = 12 m Width = 10 m Height = 9 m Explntion This shpe is ue. V = l l l 35 mm 40 mm This shpe is tringulr prism. Volume = 1 2 h l Bse of tringle = 35 mm Height of tringle = 40 mm Length = 70 mm 416 70 mm Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Exerise 11f Exmple 12 Exmple 12 1 Choose the most suitle unit (mm 3, m 3, m 3, km 3 ) for mesuring the volume of eh of these items. your penil se petrol tnker swimming pool d soft drink n e grin of rie f erel ox 2 Clulte the volume of these shpes. A = 56 m 2 5 m 3 Clulte the volume of these uoids. d 4 m 6 mm 1 m 2 m 7 mm 24 mm 5 m A = 250 mm 2 4 Copy nd omplete the tle. Length Width Height Volume 2 m 3 m 4 m 2 m 2 m 8 m 3 3 m 3 m 27 m 3 e 4 m 3 m 4 m 4 m 100 mm 4 m f 5 m A = 640 mm 2 9 m 6 m 6 m 85 mm 12 m 6 m 5 m 2 m 30 m 3 1 m 4 m 20 m 3 Chpter 11 Mesurement II 417
Exmple 13 5 Clulte the volume of these tringulr prisms. (These tringulr prisms re not to sle.) d 3 m 30 mm 6 m 20 mm 18 mm 4 m 6 Copy nd omplete the tle. e 25 mm 6 m 9 m 25 mm 11 m 25 mm 8.1 m 10.2 m 15.6 m Ojet Bse Height Length Estimte volume Clulted volume Text ook Your lssroom Box 1 Box 2 Note: Box 1 nd 2 ould e ny different ox tht is ville from your lssroom. 7 A 1 m y 1 m y 1m retngulr ox nd 2 m y 1 m y 0.5 m retngulr ox oth hve volume of extly 1 ui metre. Give the dimensions of three other retngulr oxes tht lso hve volume of 1 ui metre. 8 Effie nd Helen re shring retngulr in of length 2.8 m, width 3.2 m nd height 3 m. Wht is the volume of the in? 9 A ship hs retngulr storge re tht is 15 m long, 10 m wide nd 2.8 m high. Wht is the volume of the storge re? If ox hs the dimensions 1 m y 0.5 m y 2.5 m, wht is the volume? How mny oxes n fit into the storge re? 418 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
10 Clulte the volume of eh of the uoids desried in this tle. Length Width Height Volume 3 m 2 m 4 m 6 m 4 m 8 m 12 m 8 m 16 m 24 m 16 m 23 m How does douling the length, width nd height ffet the volume otined? 11 Drw uoid with the volume of 36 m 3. Drw different uoid with the volume of 36 m 3. Drw nother different uoid with the volume of 36 m 3. ENRICHMENT: Converting metres ue to entimetres ue 12 Clulte the volume of this ue. Clulte the volume of this ue. 1 m 1 m 1 m 100 m 100 m 100 m Wht is the sme nd wht is different out the ues in questions nd? d Using your nswers from questions nd omplete this sttement:1 m 3 = m 3 e If uoid hs volume of 3 m 3, wht is its volume in entimetre ues? f Joseph hs pddling pool tht hs volume of 0.9 m 3. Wht is its volume in ui entimetres? g The hose runs t 250 m 3 per seond. How long will it tke Joseph to fill the pddling pool? Chpter 11 Mesurement II 419
11-7 Prtil mesurement Hrith nd Lilly re orgnising the guess the numer of jellyens for the shool Pets Dy. They hve ontiner with volume of 1000 m 3 nd re trying to deide how mny jellyens will fit in the jr. Hrith suggests lulting the volume of one jellyen nd the volume of the jr. They mesure the dimensions of the jr nd estimte the mesurements of the jellyen. Hrith estimtes the volume of the jellyen to e 0.5 m 3. He then estimtes tht the numer of jellyens in the jr is 2000. Srh disgrees nd sys there must e lot less. Investigte nother wy Hrith ould estimte the numer of jellyens tht n fit in the jr. Key ides Mesurements need to e urte. Mesuring equipment used inludes: length: ruler, tpe mesure mss: kithen sles, throom sles pity nd volume: ylinders, mesuring jugs. Estimtes re not urte. They re pproximte mesurements. Exmple 12 EXAMPLE 14 Wht is the height of one illion dollrs worth of $20 notes? Solution Explntion 0.8 m 0.5 m The informtion tht we require is the thikness of $20 note nd how mny $20 notes mke one illion. Equipment required is ruler. 420 1 m Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Ten $20 notes is 1 mm Numer of $20 notes: 1 000 000 000 20 = 50 000 000 Height 50000000 10 = 5000000 mm 5000000 mm = 5000 m 5000 m = 5 km Height of $1 000 000 000 of $20 notes is 5 km Exerise 11g 1 Estimte first nd then urtely mesure these items. Copy nd omplete the tle nd lulte the differenes. Ojet My height Height of door Width of white ord Mss of pen Volume of up My estimte It would e very diffiult to mesure one $20 note. So we will mesure ten $20 notes. Using ruler mesure the height of $20 notes. There re two different definitions of illion. The short form is one thousnd million (1000 000 000) nd the long form is one million million (1000 000 000 000). We will use the short form. Note this is n estimte nd theoretil nswer. In relity this would not e possile. Friend s estimte Atul mesurement Mine Differene Friend Who ws more urte t estimting these mesurements? Disuss eh other s strtegies. 2 Investigte these four quntities nd rnk them in order from iggest to smllest. Show your working. Volume of wter to fill th Volume of wter used to wter the grden for 15 minutes Volume of wter used in wshing mhine to wsh full lod of wshing d Volume of wter used to wsh r 3 Investigte how to nswer this question. How mny mrshmllows would it tke to fill your lssroom? Wht equipment will you need? Approximtely wht is the volume of mrshmllow? How mny mrshmllows would fit into 1000 m 3? d How mny mrshmllows would fit into 1 m 3? e How mny 1 m 3 re there in your lssroom? f Estimte how mny mrshmllows would fit into your lssroom. g If you were going to do this question gin, wht would you do differently? Chpter 11 Mesurement II 421
4 Investigte how to nswer this question. How mny wter lloons would it tke to fill your lssroom? Wht equipment will you need? Approximtely how muh wter fits into wter lloon? How mny filled wter lloons would fit into 1 m 3? d How mny 1 m 3 re there in your lssroom? e Estimte how mny wter lloons would fit into your lssroom. f If you were going to do this question gin, wht would you do differently? 5 Investigte how to nswer this question. How long will it tke to fill pddling pool? Wht equipment will you need? Approximtely how muh wter flows out of tp in 10 seonds? Wht is the volume of the pddling pool? d If 1 m 3 is the sme s 1 ml of wter, how mny millilitres of wter is required to fill the pddling pool? e Estimte how long it will tke to fill the pddling pool. f If you were going to do this question gin, wht would you do differently? 6 Investigte how to nswer this question. How mny hirs re on your hed? Wht equipment will you need? How mny hirs re in pproximtely 1 m 2 on your hed? Estimte the re of the hed tht is overed in hir. d Estimte how mny hirs re on your hed. 7 Investigte how to nswer this question. Wht is the length of one million $2-oin tril? Wht equipment will you need? Wht is the dimeter of $2-oin? How mny $2 oins re required? d Wht is the length of the oin tril in m? e Wht is the length of the oin tril in km? 125 m 50 m 422 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
Thinking mthemtilly Volume nd pity of irregulr shpes For this investigtion, you will need: three smll, different-sized irregulr solids (for exmple, stone, die) ontiner nd try (the ontiner must fit into the try) mesuring jug tht n mesure millilitres. You will need to work in group s direted y your teher. Estimting nd mesuring the volume of retngulr solids 1 i Estimte the volume of eh irregulr solid. iii List the irregulr ojets nd their volumes in order, from lrgest to smllest. Mesure the dimensions of eh irregulr ojet nd determine their tul volume. Were your estimtes lose to the tul volumes? 2 Find the re of eh of the following retngles: 1 4 m 1 3 m 1 2 m 2 m 3 m 2 m 4 m 1 2 m 1 m 4 m 1 4 m 3 m 1 3 m Clulting wter displement 2 Ple the ontiner in the try nd fill the ontiner to the rim with wter. Crefully lower one of the retngulr ojets into the wter (e reful not to put your fingers in the wter). Collet the wter tht spills over nd pour it into the mesuring jug. Reord how mny millilitres of wter were displed y the ojet. Repet prts nd for the other two retngulr ojets. Refleting Ptterns from squre metres Finding res 1 This digrm shows squre of re 1 squre metre. Drw two other retngles tht hve n re of 1 squre metre. Compre the wter displed y eh retngulr ojet with the volumes you lulted ove. Write prgrph explining wht you notie. 1 m 1 m Reord their dimensions nd re in tle. 3 Wht do you notie out the re of eh shpe? 4 Is 1 squre metre different from 1 metre squred? Chpter 11 Mesurement II 423
11 Chpter summry The metri system The metri system is used to mesure mss, length nd volume. Length is mesured in millimetres, metres nd kilometres. 10 mm = 1 m 100 m = 1 m 1000 m = 1 km Mss is mesured in milligrms, grms, kilogrms nd tonnes. 1000 mg = 1 g 1000 g = 1 kg 1000 kg = 1 t Cpity is mesured in millilitres, litres, kilolitres nd meglitres. 1000 ml = 1 L 1000 L = 1 kl 1000 kl = 1 ML Perimeter of polygons To find the perimeter of ny polygon you dd the lengths of ll the sides. The mesurements must e expressed in the sme unit efore they n e dded. The units of perimeter re millimetres, entimetres, metres nd kilometres. It is importnt to write the unit of the nswer. 8 n men mny different things, ut 8 m gives more informtion. Perimeter of irles Prts of irle: rdius, dimeter, irumferene C i r u m f e r e n e = π d i m e t e r C = π d Cirumferene = π two lots of r d i u s C = 2 π r Are of polygons Are is the spe tht two-dimensionl ojet tkes up. Units for re: millimetres squred (mm 2 ), entimetres squred (m 2 ), metres squred (m 2 ), kilometres squred (km 2 ) Lnd re is mesured in hetres (h). 10 000 m 2 = 1 h. dimeter rdius irumferene 424 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9
The re of ommonly used polygons re: length (l) Are = length width Are = l w width (w) Are of irles nd ompound shpes Are of irle = π r 2 To lulte the re of ompounded shpe, divide the shpe into retngles, tringles or semi-irles. se () height (h) Are = se height Are = h To lulte the re of the shded region sutrt the re of the shpes. Volume of prisms Volume is mesure of the mount of spe three-dimensionl shpe oupies. Cpity is the mount of something n ojet n hold. se () height (h) Are = 1 2 se height Are = 1 2 h The most ommon mesures of volume re: ui metres (m 3 ) ui entimetres (m 3 ) ui millimetres (mm 3 ) To lulte the volume of ny three-dimensionl shpe V = re of se height. Volume of two ommon three-dimensionl shpes. height Cuoid length Prtil mesurement width width Tringulr prism length height Volume = l w h Volume = 1 2 l h w Mesurements need to e urte. Mesuring equipment used inludes: length: ruler, tpe mesure mss: kithen sles, throom sles pity nd volume: ylinders, mesuring jugs. = + Estimtes re not urte. They re pproximte mesurements. = = + Chpter 11 Mesurement II 425
Review questions 1 Clulte the perimeter of these shpes. 8.8 m 3 The ost of fening wire is $13 per 100 m. How muh will Sestien s fening wire ost? Clulte the perimeter of these shpes. 28.2 m 89 mm 4.7 m PL E 195 mm 7.3 m 4 74 mm Clulte the re of these shpes. 68 mm 18 m 4 m 52 mm SA M 123 mm d 43 mm 2 60 mm 45 mm 5 Sestien is uilding fene for retngulr pddok. The fene required six rows of wire. Clulte the re of these shpes. 5 m 18 m 73 m 14 m 52 m Clulte the perimeter of the pddok. If there is only 2 m gte to enter the pddok, how muh wire will e required to uild the fene? 24 m 65 mm 72 mm 426 Mthemtis & Sttistis for the New Zelnd Curriulum: Yer 9 978110762840311_p384-427.indd 426 09/08/13 11:37 PM
d 25 mm 10 mm 6 Clulte the volume of these shpes. 36 mm 320 m 2 18 m 15 m 54 mm 16 m 23 m 41 mm 7 Rnk these quntities in order of iggest to smllest. Distne of mrthon Distne from plnet Erth to the Moon Height of Mt Trnki d Distne from Auklnd to Inverrgill 8 Sestien onstruted shpe out of reyled oxes. 6 m 6 m 5 m 6 m 6 m 3 m 24 m 18 m Clulte the totl volume of the onstrution. 9 Convert these mesurements. 450 m = m 5.8 L = ml 0.083 kg = g d 0.91 t = kg e 650 mm = m f 72 L = ml 10 A retngulr pool is 6 m 7.5 m, nd is surrounded y pth 1 m wide. Wht is the re of the pool? Wht is the re of the pth? If the pool is 1.5 m deep nd is filled with wter to 20 m from the top, how muh wter is in the pool? d If the pool is filled t the rte of 40 litres per minute, how long will it tke to fill the pool to 20 m from the top? 11 A retngulr fmily room is 6 m 4 m. Fye is onsidering rpeting the room with either 2 m wide rpet osting $120 per liner metre or rpet tiles osting $65 per squre metre. Wht is the re of the room? Wht is the ost of using the rpet squres? How mny metres of the 2 m wide rpet is needed to rpet the fmily room? d Wht is the ost of using the 2 m wide rpet? e Whih option is more expensive nd y how muh? Chpter 11 Mesurement II 427