8 Measurement. How is measurement used in your home? 8E Area of a circle 8B Circumference of a circle. 8A Length and perimeter

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1 8A Length nd perimeter 8E Are of irle 8B Cirumferene of irle 8F Surfe re 8C Are of retngles nd tringles 8G Volume of prisms 8D Are of other qudrilterls 8H Are nd volume onversions SA M PL E Mesurement nd geometry 8 Mesurement E ss e n ti l Qu e s tion How is mesurement used in your home? 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:07 PM

2 Are you redy? 8A 8A 8F 1 How mny millimetres long is 15-m ruler? 2 Wht is the perimeter of this shpe? does this ojet hve? 4 m A 1 6 How mny fes B 1800 m 8F 7 Whih net mthes the ojet in question 6? A 3 m C 20 m D 23 m 8C 3 Wht is the re of this shpe? 4 Wht is the re of PL this tringle? A 24 m2 B 2 M C 10 m2 SA D 2 m2 8D B E 8C 3 m 5 Whih shpe is rhomus? A C D 4 m 8G 8 Wht is the volume of this prism? A 3 B B 13 C 13 D 30 m3 C D 8H 9 How n you onvert entimetres to metres? A divide y 100 B multiply y 100 C multiply y 10 D divide y 10 8H 10 Whih unit reltes to re? A mm3 B m m3 D km2 C 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:07 PM

3 402 CHAPTER 8: MEASUREMENT 8A Length nd perimeter Strt thinking! When you think out mesurement, two words you my know re length nd perimeter. 1 Wht is the differene etween length nd perimeter? Common units of length re millimetres (mm), entimetres (m), metres (m) nd kilometres (km). 2 Think of n ojet tht you ould mesure oth the length nd perimeter of in: i millimetres ii entimetres iii metres iv kilometres. 3 With lssmte, estimte: the length the perimeter of the tle tht you re sitting t now. 4 Why is it importnt to estimte? 5 Choose devie to mesure the tle. Explin why your hosen devie is etter thn either 5-m ruler or trundle wheel. 6 Imgine your tle is ginst wll. How might you find the perimeter of the tle without mesuring the side tht is ginst the wll? 7 Without moving the tle, find oth its length nd its perimeter. How do these mesurements ompre to your estimte? Complete 8A Disover tsk to lern more. Key ides Common units of length re millimetres (mm), entimetres (m), metres (m) nd kilometres (km) You n onvert etween units of length y multiplying or dividing y the onversion ftor. km m m mm Perimeter is the distne round the outside edge of 2D (two-dimensionl) shpe To lulte perimeter, dd the length mesurements of eh side together. Mke sure tht ll mesurements re in the sme unit. Alwys estimte your nswer first

4 8A Length nd perimeter 403 Exerise 8A Length nd perimeter A <numers to ome> B <numers to ome> C <numers to ome> Exmple 8A-1 Converting units of length Convert 1.23 m into millimetres. Think 1 First onvert to entimetres y multiplying y m = 100 m. 2 Complete the onversion to millimetres y multiplying y m = 10 mm. Write 1.23 m = ( ) m = 123 m = (123 10) mm = 1230 mm understnding AND FluenY 1 Convert: 58 m into entimetres 2.7 km into metres 521 m into metres d 398 m into kilometres e 1.2 m into millimetres f 3987 m into kilometres g 423 mm into entimetres h 7 mm into entimetres 2 Convert: 5.34 m into millimetres 3.95 km into entimetres mm into metres d mm into kilometres e 236 into kilometres f 287 mm into metres g km into entimetres h 0.91 km into millimetres

5 404 CHAPTER 8: MEASUREMENT Exmple 8A-2 Clulting perimeter Clulte the perimeter of this shpe in entimetres. 9 m 111 mm 78 mm 42 mm 21 m Think 1 There re six sides to the shpe so there re six lengths to e dded. 2 Write ll mesurements in entimetres. Rememer tht 10 mm = 1 m. Write perimeter = 9 m mm + 78 mm + 42 mm + 21 m + 3 Add ll the lengths nd inlude the orret unit. = 65.1 m = 9 m m m + 21 m + understnding AND FluenY 3 Clulte the perimeter of eh shpe. 3 m 3 m 2 m d 2 m 2 m 2 m e 41 mm f mm 11 m 4 m 4 m 59 mm 59 mm 5.2 m 25 mm 3 m 30 mm 55 mm 4.4 m 7.9 m 4.1 m 30 mm 15.1 m 4 For eh ojet, first estimte nd then mesure: i the horizontl nd vertil lengths ii the perimeter. 5 Explin how you found the perimeters in question 4.

6 8A Length nd perimeter 405 understnding AND FluenY Prolem solving nd resoning 6 Nme the est unit to use to mesure: the length of sketll ourt the perimeter of n ipod 7 Nme devie you ould use to mesure eh ojet in question 6. 8 Clulte the perimeter of eh ojet. 50 mm e 3 m 4 m 2 m 2 m 1 the perimeter of Austrli d the length of your textook. d f 9 m 11 mm 4 m 13 m 19 m 3 mm 22 mm 2 mm 7 mm 9 Clulte the perimeter of: squre with side length 4 m squre with side length 1.2 km n equilterl tringle with side length 2. d rhomus with se 15 mm. e retngle with length 10 mm nd width 15 mm f retngle with length 6 m nd width 1 m m 10 To wrm up efore netll mth, Jordn hs the option of either running six lps round the netll ourt or four m shuttle runs. How fr is one lp round the netll ourt? How fr is six lps round the netll ourt? A shuttle run involves strting t the ottom of the ourt nd running hrd to the first third-line, k to the strt, running to the seond third-line nd k to the strt nd finlly up to the end of the ourt nd k. How fr is the first prt of the shuttle run (to the first third-line nd k)? d Wht distne does single shuttle run over? e How fr is four shuttle runs? f If you were Jordn, whih wrm up tivity would you hoose? Explin. Note The dimensions of the ourt re 30.5 m y m or 30.5 m m.

7 406 C HAPT E R 8 : M E A S U R E M E NT P r o l e m s o lv i n g n d r e s o n i n g 11 Alie wnts to mke some photo frmes for 1 m friend. She uys some wood to fit round the outside of stndrd-sized photo. Wht is the perimeter of stndrd-sized photo? Alie wnts the frme to e 1 m wide, s shown in the imge. 1 Why is the length of wood needed greter thn the perimeter of the photo? (Hint: re the orner prts of the frme inluded in the photo perimeter?) Wht length of wood is needed? d Wht is the perimeter round the photo frme? Why is this different from prt? e Investigte how these results would hnge if: i the frme ws 2 m wide ii the frme ws 3 m wide E iii the frme ws for 1 20 m photo PL iv the frme ws for 20 m 2 photo. Write prgrph on your findings. M 12 Alie pks the photo frmes from question 11 SA into gift ox with the dimensions shown in the photo nd wnts to tie rion round the present. Assuming tht she will need minimum of 60 m of rion for the ow, how muh rion will she need to uy in order to wrp the present s shown? 1 20 m 20 m 13 A squre hs perimeter of 3. Wht is its side length? 14 Give three possile sets of dimensions for retngle with perimeter 100 m. 15 A retngle hs perimeter 60 m. Wht might its dimensions e? How mny nswers re possile for prt? Assume whole numers only. 16 Mke up your own question like question 15 nd give it to lssmte to solve. How mny possile whole-numer nswers re there? 17 A new television sreen hs perimeter of 6.1 m. If it hs length of 1.9 m, find its width. 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:07 PM

8 8A Length nd perimeter 407 Chllenge 18 A retngulr vegetle grden hs perimeter of 28 m. Write down possile set of dimensions for this grden. A fene is to e pled 1 m from the edge round the vegetle grden to llow spe for more vegetles. Wht length of fening is required? Wht length of fening would e required if the distne etween the fening nd the edge of the grden ws: i 2 m? ii 3 m? d Cn you see pttern or rule tht you ould follow in order to find the length of fening required, no mtter how lose or fr from the edge it is pled? e Repet prts d using different set of dimensions. Do your findings mth the rule you found in prt d? Explin. Reflet How is estimtion importnt when mesuring length nd perimeter?

9 408 CHAPTER 8: MEASUREMENT 8B Cirumferene of irle Strt thinking! 1 How would you mesure the perimeter of irle? The perimeter of irle is lled its irumferene. 2 Pir up with lssmte nd get ruler nd long piee of string or rion. 3 Copy this tle, leving spe for t lest five rows. Ojet Dimeter (D) Rdius (r) Cirumferene (C) C D 4 Either s lss or in your pirs ollet t lest five irulr ojets. Cylinders will work s well. 5 Choose one ojet. Mesure ross the width of the irle t its entre s shown here. This is lled its dimeter. Reord this in your tle. Be s urte s possile. The rdius is the distne from the entre of the irle to its perimeter. It is extly hlf the dimeter. 6 Wht is the rdius of the irulr ojet you re mesuring? Reord this in your tle. 7 Use your piee of string nd ruler to mesure the irumferene of the irulr ojet nd reord the mesurement in your tle. 8 Complete the lst ell of your tle, orret to two deiml ples. Wht numer do you otin? 9 Repet steps 5 8 for ll your other irulr ojets. Wht do you find? The rtio of the irumferene to the dimeter ( C ) is equl to the irrtionl numer pi (π), no mtter D wht size irle is. Rememer tht π nnot e written s n ext deiml numer. Rounded to two deiml ples, the vlue of π is pproximtely Why might the numer you get eh time not e lose to π? Complete 8B Disover tsk to lern more. Key ides Cirumferene (C) is the perimeter of irle. Dimeter (D) is the width of irle through its entre. Rdius (r) is hlf the dimeter. To find the irumferene of irle, multiply the dimeter y π. C = πd or C = 2πr. irumferene dimeter rdius

10 8B Cirumferene of irle 409 Exerise 8B Cirumferene of irle A <numers to ome> B <numers to ome> C <numers to ome> For ll questions in this topi, use π on your lultor nd then round eh nswer to two deiml ples. Exmple 8B-1 Finding dimeter Find the length of the dimeter if irle hs rdius of 4 m. understnding AND FluenY Think 1 The dimeter is twie the length of the rdius. Multiply the rdius y two. Write D = 2r = 2 4 m 2 Write your nswer. = 1 Find the length of the dimeter of eh irle. 2 mm 2 Find the length of the dimeter if irle hs rdius of: 3 m 6 m 70 mm d 1. e 111 mm f 9 m g 12 mm h 2.5 m i 25 mm Exmple 8B-2 Finding rdius Find the length of the rdius if irle hs dimeter of. Think Write 1 The rdius is hlf the size of the dimeter. Divide the dimeter y two. r = D 2 = 2 2 Write your nswer. =

11 410 CHAPTER 8: MEASUREMENT understnding AND FluenY 3 Find the length of the rdius of eh irle. 12 mm 4 Find the length of the rdius if irle hs dimeter of: 4 m 6 mm d 30 m e 5.6 mm f 9 m g 1 h 5 m i 24 m 7 m Exmple 8B-3 Using the dimeter to lulte irumferene Clulte the irumferene of this irle, orret to two deiml ples. Think Write 1 Identify whih formul to use. C = πd 2 Sustitute the mesurement for dimeter. = π 6 3 Clulte using π on your lultor. = Round to two deiml ples nd inlude the pproprite unit. C = Clulte the irumferene of eh irle. 2 mm 7. d e f 15 mm 1.8 m 42 mm

12 8 B C i r umf er en e o f irl e Exmple 8B Using the rdius to lulte irumferene Clulte the irumferene of this irle, orret to two deiml ples. Think Write 1 Identify whih formul to use. C = 2πr 2 Sustitute the mesurement for rdius. =2 π 5 3 Clulte using π on your lultor. = Round to two deiml ples nd inlude the pproprite unit. C = m E PL M m 21 m SA u n de r s t n ding A N D F lu e n Y 6 Clulte the irumferene of eh irle. d e 9 mm f m 7 Clulte the irumferene of eh ojet. d 26 mm 13 m 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:07 PM

13 412 CHAPTER 8: MEASUREMENT understnding AND FluenY 8 Find the irumferene of irle with: rdius 1.2 m dimeter 2. dimeter 6.4 mm d rdius 0.4 m e rdius 19 m f rdius 13 mm g dimeter 0.75 m h dimeter 2. 9 Look t this semiirle. How muh of irle is it? Wht would e the irumferene if it ws full irle? Use your nswers to prts nd to find the length of the urved setion of the semiirle. d Wht is the perimeter of the semiirle? 3 m 10 Use your understnding of irumferene to find the perimeter of eh shpe. Prolem solving nd resoning 2 m 4 m d e f 10 mm 25 mm m Elli ought Thoms ring. She knows tht his ring finger hs irumferene of 60 mm. She uys him n S size ring, whih hs n inner dimeter of mm. Find the inner irumferene of this ring. Do you think it will fit? Elli gives the ring to Thoms nd finds tht it doesn t fit euse it won t go over his knukle. She tries oth T size ring (dimeter of mm), whih is still slightly too smll, nd U size ring (dimeter of mm), whih is slightly too ig. Wht size rnge must the irumferene of Thoms knukle e in? (Hint: find the irumferenes of oth the T nd U size rings.) Elli disovers tht these rings lso ome in hlf sizes. Wht would e the dimeter of ring size T 1 2? d If the ring is perfet fit, find the irumferene of Thoms knukle. 12 Jk goes for ike ride with his younger sister Lily, ut she hs troule keeping up. Jk gets frustrted ut his mother explins tht euse Lily s ike is smller, Lily hs to work hrder to go the sme distne. If Jk s ike wheels hve dimeter of 5 nd Lily s hve dimeter of 32 m: find the irumferene of their wheels find how mny times eh wheel turns to trvel 1 km disuss whether Jk s mum is orret.

14 8 B C i r umf er en e o f ir l e 413 P r o l e m s o lv i n g n d r e s o n i n g 13 Luren nd Mtt go for ride on merry-go-round. Luren hooses horse 5.4 m from the entre nd Mtt hooses horse 4.1 m from the entre. Do these lengths relte to dimeter or rdius? Explin. Find how fr they will eh trvel in one revolution of the merry-go-round. If the merry-go-round rottes six times in ride, find the differene in the distne Mtt nd Luren trvel. d Explin why, lthough Mtt nd Luren re on the sme ride, Luren trvels further thn Mtt. 14 For irles with these irumferenes, find: i the dimeter ii the rdius. 314 m 6.28 m 15.7 mm d Imgine you ould tie piee of string round the irumferene of the Erth t the equtor. E If Erth hs dimeter t the equtor of out km, how muh string would e needed? PL If you dded 1 m to the length of the piee of string, would you e le to: i squeeze your pinkie finger under the string? ii fit your hnd under the string? M iii slide yourself under the piee of string? SA iv rwl under the string? v wlk under the string? Explin your nswer. Chllenge 16 A lol show hs new Ferris wheel. It hs rdius of 54 m. Wht is the dimeter of the Ferris wheel? If the Ferris wheel stnds 5 m off the ground, how high would you e if you were t the top? Wht is the irumferene of the Ferris wheel? d How mny rriges ould fit on the Ferris wheel if eh one ws 1 m wide nd there hs to e t lest 3 m etween eh rrige? e Use your nswer from prt to lulte the distne etween eh rrige on the Ferris wheel. f If the owner wnted to fit 100 rriges onto the Ferris wheel, ssuming minimum distne of 3 m etween eh rrige, y how muh would the rdius of the Ferris wheel hve to inrese? g Are the detils in this question relisti? Investigte nd write prgrph on your findings. 08_NOL_OM_SB8_29418_TXT_2pp.indd 413 Reflet Wht is π nd how does it relte to rdius, dimeter nd irumferene? 20/01/14 10:07 PM

15 414 CHAPTER 8: MEASUREMENT 8C Are of retngles nd tringles Strt thinking! 1 Wht is re nd how is it different from length? 2 For the retngle in figure A, wht is: the length (l)? the width (w)? the re (A)? 3 How does figure A show tht the formul for the re of retngle is A = l w or A = lw? Figure A Figure B shows the retngle from figure A split into two identil tringles. 4 How does figure B show tht the re of tringle is extly hlf tht of retngle with the sme dimensions? Key ides Figure B 5 How do the se () nd height (h) of this tringle relte to the length nd width of this retngle? 6 The formul for the re of tringle is A = 1 2 h or A = 1 2 h. How does this relte to the formul for the re of retngle? Complete 8C Disover tsk to lern more. Are (A) is the mount of spe enlosed y 2D shpe. Common units of re mesurement inlude squre millimetres (mm 2 ), squre entimetres (m 2 ), squre metres (m 2 ) nd squre kilometres (km 2 ). To find the re of retngle, multiply its length (l) y its width (w). A = lw l w To find the re of tringle, multiply its se () y its height (h) then divide y 2. A = 1 2 h Note The se nd height of tringle must e perpendiulr to eh other. h

16 8C Are of retngles nd tringles 415 Exerise 8C Are of retngles nd tringles A <numers to ome> B <numers to ome> C <numers to ome> Exmple 8C-1 Clulting re of retngle Clulte the re of this retngle. 2 m 45 mm understnding AND FluenY Think Write 1 Write the formul. A = l w 2 Identify l nd w. Write the length in entimetres so tht oth l nd w re in the sme units. 1 Clulte the re of eh retngle. 10 mm 7 mm 8 d 3 m 3 mm l = 45 mm = 4. w = 2 m 3 Sustitute the vlues for l nd w into the formul. A = Clulte the result nd inlude the pproprite unit. = 9 m 2 50 mm 9.8 mm 1.2 m e 1.1 m f 7 mm 90 m 2.3 m

17 416 CHAPTER 8: MEASUREMENT Exmple 8C-2 Clulting re of tringle Clulte the re of this tringle mm Think Write 1 Write the formul. A = 1 2 h 2 Identify nd h. Write the se in entimetres so tht oth nd h re in the sme units. = 80 mm = h = 1. 3 Sustitute the vlues for nd h into the formul. A = Clulte the result nd inlude the pproprite unit. = 2 understnding AND FluenY 2 Clulte the re of eh tringle. 9 m d 36 mm m 21 m 120 m e 13 f m 3.8 m 3 Clulte the re of eh shpe. retngle with length 35 mm, width 4 m retngle with length 12 mm, width 1 tringle with se 15 mm, height 2 m d squre with length 7.2 mm e tringle with se 2, height 10 mm f tringle with se, height 42 mm g squre with se 19 m

18 8C Are of retngles nd tringles 417 h retngle with se 4.2 m, height 11 mm understnding AND FluenY i tringle with se 0.7 m, height 49 m 4 How muh mteril is needed to mke the multi-oloured umrell shown? 5 A lssmte ttempted question 4 nd fter using 68. s the height of eh tringle, otined n nswer of Cn you explin where they went wrong? Clulte the re of eh shpe. e 4 m 20 mm 45 mm 45 mm 20 mm 1 1 d f 4 m 3 m 7 Two lssmtes were rguing out the dimensions of the retngle in question 6. Jun sid tht the length ws the longest side nd therefore it ws 45 mm. Eri sid it ws the horizontl length nd therefore it ws 20 mm. Cn you explin how they re oth wrong nd oth right? 8 You n use your understnding of res of retngles nd tringles to find the 3 m re of omposite shpes. Copy this figure nd drw line on it splitting it into two retngles. Find the re of eh retngle nd hene the re of the entire shpe. Is there nother wy you ould hve done this? Drw nother digrm to show how it ould e done. 1 m 2 m 2 m

19 418 CHAPTER 8: MEASUREMENT 9 Find the re of eh shpe. understnding AND FluenY 10 mm 9 m 8 mm 7 mm 4 mm 3 mm 12 mm 12 m 7 m 12 m d 8. 1 m m 9 m 15.6 m 18.2 m 9 m Prolem solving nd resoning e 3 m 10 Guyn s flg onsists of numerous oloured tringles. Use the mesurements to find the re of eh oloured setion. 11 This envelope templte hs n re of m 2. Use your understnding of res of retngles nd tringles to prove this re mesurement is orret. Crete your own envelope (using retngles nd tringles) tht hs totl re of: i less thn 300 m 2 ii etween 300 m 2 nd 400 m 2 iii more thn 400 m 2. Wht did you hnge eh time? f 4 m m 46.3 m m 10.2 m 10.2 m 10.2 m 56.9 m 12 A retngle hs n re of 6 2. Drw three possile retngles tht fit this desription, lelling length nd width on eh one.

20 8C Are of retngles nd tringles 419 Prolem solving nd resoning 13 A retngle hs n re of 20 m 2 nd length of 4 m. Find its width nd explin why there is only one retngle tht you n drw tht fits this desription. 14 A tringle hs n re of 32 mm 2. Drw three possile tringles tht fit this desription, lelling se nd height on eh one. Suppose tht this tringle hs se of 8 mm. Wht would its height e? Drw two possile tringles tht fit this desription, lelling se nd height on eh one. d Explin why there re mny tringles tht fit this desription. 15 A retngle hs n re of 56 m 2. Stte three possile sets of dimensions. Find the perimeter of eh possile retngle. Chllenge 16 A retngle hs perimeter of 4. Find three possile res this retngle ould hve. 17 A squre hs n re of 64 mm 2. Drw this squre, stting its dimensions, its perimeter nd why there is only one squre tht mthes this desription. 18 A wooden frme is to go round retngulr mirror with n re of 3600 m 2. If the mirror hs length of 90 m, wht is its width? Wht re of wood is required if the frme is to e wide? (Hint: drw piture to help you with your lultions.) 19 Mohmmed is mking retngulr vegetle grden nd hs 24 m of fening to put round the grden. List three possile sets of dimensions nd orresponding res for the grden. Wht is the mximum re tht you n find using 24 m of fening? Wht shpe is this? Mohmmed relises he n enlose igger re if he puts the grden ginst the shed wll nd only hs to fene three sides of the grden. List nother three possile sets of dimensions nd orresponding res for the veggie grden if one side is ginst the shed wll. d Wht is the mximum re tht you n find now using 24 m of fening for three sides? Wht shpe is this? How does this relte to your nswer to prt? Reflet How re the dimensions nd re of tringle relted to the dimensions nd re of retngle?

21 420 CHAPTER 8: MEASUREMENT 8D Are of other qudrilterls Strt thinking! You n use your understnding of the re of retngle to find the re of other qudrilterls. 1 Mth eh desription to prllelogrm, rhomus or kite. Opposite sides prllel, ll sides equl, opposite ngles equl. Two pirs of sides equl in length. Opposite sides prllel nd equl in length, opposite ngles equl in size. 2 Look t the prllelogrm shown in figure A. h Wht do nd h stnd for? How might you ut nd rerrnge this prllelogrm Figure A so it forms retngle? Use the formul for re of retngle to show the formul for re of prllelogrm is A = h or A = h. 3 Look t the kite in figure B. The digonls re mrked inside the kite nd lelled x nd y. Figure B Copy nd ut out the kite twie. Cut one of these opies long its digonl lines. Rerrnge ll the prts so tht you form retngle. How do the digonls of the kite relte to the length nd width of the retngle? d How mny opies of the kite mke up this retngle? e Use the formul for the re of retngle to show tht the formul for the re of kite is A = 1 2 x y or A = 1 2 xy. Complete 8D Disover tsk to lern more. x y y x Key ides The formul for the re of prllelogrm is A = h h The formul for the re of kite is A = 1 2 xy x y The formul for the re of rhomus is A = h or A = 1 2 xy h y x

22 8D Are of other qudrilterls 421 Exerise 8D Are of other qudrilterls A <numers to ome> B <numers to ome> C <numers to ome> Exmple 8D-1 Clulting re of prllelogrm Clulte the re of this prllelogrm. 9 m Think Write 1 Write the formul. A = h 2 Identify nd h. =, h = 9 m 3 Sustitute the vlues for nd h into the formul. A = Clulte the result nd inlude the pproprite unit. = Clulte the re of eh prllelogrm. 9 mm 3 m 7 mm 14 m Exmple 8D-2 Clulte the re of this kite. Clulting re of kite 9 m Think Write 1 Write the formul. A = 1 2 xy 2 Identify x nd y. x =, y = 9 m 3 Sustitute the vlues for x nd y into the formul. A = Clulte the result nd inlude the pproprite unit. = 2 2

23 422 CHAPTER 8: MEASUREMENT 2 Clulte the re of eh kite. 8 mm 4 m 12 mm 3 m 4 m Exmple 8D-3 Clulting re of rhomus Clulte the re of this rhomus. 11 mm understnding AND FluenY Think 1 A rhomus is oth prllelogrm nd kite. As the se nd height re given, write the formul for prllelogrm. Write A = h 10 mm 2 Identify nd h. = 10 mm, h = 11 mm 3 Sustitute the vlues for nd h into the formul. A = Clulte the result nd inlude the pproprite unit. = 110 mm 2 3 Clulte the re of eh rhomus. 5 mm 3 mm 4 mm 4 For eh shpe: i identify if it is prllelogrm, rhomus or kite ii write the formul to find its re iii lulte its re. 2 m 3 m 9 mm 1

24 8D Are of other qudrilterls 423 understnding AND FluenY Prolem solving nd resoning d e f 4 mm 9 mm 7 mm 16 mm 5 Clulte the re of eh qudrilterl. 3 m 2 m e 90 mm 11 m 11 m 9 m d f 9 m 16 mm 7 mm 25 mm 13 m 9 mm 4 m 7 mm 6 Tim s unle hs n old rod sign he wnts pinted on oth sides. If the sign is 0.7 m tll nd 0.7 m wide, find the totl re Tim will hve to pint if he uses three ots of pint. Tim s unle likes the results so muh he sks Tim to pint nother 20 rod signs. Wht is the totl re Tim will pint for 20 rod signs? Rememer they need three ots on oth sides. How muh will it ost to pint the signs if litre of pint osts $8.99 nd overs 15 m 2? d If Tim is pid $150, find how muh money Tim will mke fter deduting the ost of the pint. 10 mm

25 424 C HAPT E R 8 : M E A S U R E M E NT P r o l e m s o lv i n g n d r e s o n i n g 7 Amnd wnts to mke some kites. She deides to mke two types, lrge lue kite nd smller red kite. The lue kite needs to e t most 9 long nd 6 wide. Wht is the lrgest re it n e? The red kite n hve mximum re of 400 m2. If the kite is 40 m long, how wide must it e? 8 The re of kite n e found y relting it to prllelogrm. Drw kite, using dotted lines for the digonls. Cut the kite in hlf long one of the digonls. Rerrnge these two piees so tht they form prllelogrm. d Wht is the formul for the re of prllelogrm? e One of the digonls reltes to the se of the prllelogrm. Whih one? f Hlf of one of the digonls reltes to the height of the prllelogrm. Whih one? 1 g Explin how this method lso gives the formul for the re of kite s A = 2 xy. E 9 Consider this uilding in Wht shpe is it? M If the glss setion is 21 m tll nd 86 m long, wht is the re of this entire setion? PL Hmurg. SA Is every window equl in re? Explin. d How might you lulte the re of every window? Explin. 10 Use the Internet or nother resoure to investigte the use of qudrilterls, espeilly prllelogrms, rhomuses nd kites, in rt nd rhiteture. 11 The re of prllelogrm is 40 m2. Drw three possile prllelogrms tht fit this desription, lelling se nd height. Why is this more diffiult thn if it ws retngle? 12 A rhomus hs n re of 24 mm2. Drw two possile rhomuses tht fit this desription, lelling se nd height. Find the perimeter of these two rhomuses. Explin why the re of rhomus n never e squre numer unless the shpe is squre (rememer tht squre is speil type of rhomus). 13 Drw three possile kites tht hve n re of 36 m2, lelling the pproprite dimensions. 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:07 PM

26 8 D A r e o f ot h er q u d rilt e rls Write your own open-ended questions out the res of qudrilterls nd swp them with lssmte. Wht strtegies n you use to solve these prolems? 15 The speil qudrilterl tht hs not een Chllenge overed yet is the trpezium. Look t the trpezium in figure A. h Copy nd ut out this trpezium twie. Lel the opies s shown in figure A. Figure A Rotte one of these trpeziums nd join it up to the first trpezium s shown in figure B. h Wht shpe does this now form? d Wht is the formul for the re of prllelogrm? Figure B e The height of the trpeziums orresponds to the height of the prllelogrm. Wht prt of the prllelogrm does the sum of the trpezium lengths ( + ) orrespond to? How mny trpeziums mke up this prllelogrm? PL f E The formul for the re of this prllelogrm is A = ( + ) h, where + is the se. g Explin how this shows tht the formul for the re of trpezium 1 1 is A = 2 ( + ) h or A = 2 ( + )h. M h If = 4, = 9 nd h = 3, find the re of the trpezium shown in figure A. 1 SA 16 Find the re of eh trpezium using the formul A = 2 ( + )h. 9 m 1 8 mm 16 mm 15 mm 11 m m 42 m 17 Find the ost of fitting the frme on the right with mirror if your lol hrdwre store hrges $0.25 per m2 for mirror glss. 18 The formul for the re of trpezium n lso e developed y utting trpezium in hlf nd joining the two piees to form nother prllelogrm. Drw trpezium nd ut it out to show how this might e done. 08_NOL_OM_SB8_29418_TXT_2pp.indd 425 Reflet How is the formul for the re of retngle useful when finding the formuls for the res of other qudrilterls? 20/01/14 10:07 PM

27 426 CHAPTER 8: MEASUREMENT 8E Are of irle Strt thinking! 1 If the irumferene of irle is C = 2πr, wht would e the formul for hlf irumferene of irle? 2 Drw irle tht hs rdius of nd shde hlf of it in olour, s shown in figure A. 3 How does your nswer to question 1 relte to figure A? A setor is slie of irle tht strts from the entre of the irle. 4 How mny setors does the slie of lime shown ove hve? 5 Divide your irle from question 2 into 16 equl setors so you hve eight Figure A oloured setors nd eight non-oloured setors. 6 Cut these up nd rrnge them s shown in figure B. 7 Wht shpe does this form? Drw the qudrilterl nd lel its se nd height. 8 Wht is the formul for the re of this Figure B shpe? 9 Mesure the se nd height of the shpe you formed nd lulte the re. 10 If the height of your prllelogrm is the sme s the rdius (r) of the irle nd the se of the prllelogrm is πr, explin why the formul for the re of irle is A = πr 2. Complete 8E Disover tsk to lern more. Key ides The formul for the re of irle with rdius r is A = πr 2. If you re given the dimeter, find the rdius efore using the formul. r

28 8 E A r e o f irl e 427 Exerise 8E Are of irle A B <numers to ome> <numers to ome> C <numers to ome> For ll questions in this topi, use π on your lultor nd then round eh nswer to two deiml ples. Exmple 8E-1 Clulting re of irle, given rdius Clulte the re of this irle. Write E Think PL 1 Write the formul. A = πr2 = π 72 3 Clulte the result using π on your lultor. = M 2 Identify r nd sustitute it into the formul. A = m2 SA 4 Round to two deiml ples nd inlude the pproprite unit. u n de r s t n ding A N D F lu e n Y 08_NOL_OM_SB8_29418_TXT_2pp.indd Clulte the re of eh irle. 1 9 mm 20/01/14 10:07 PM

29 428 CHAPTER 8: MEASUREMENT Exmple 8E-2 Clulting re of irle, given dimeter Clulte the re of this irle. 8 m understnding AND FluenY Think Write 1 Write the formul. A = πr 2 2 Identify the rdius (hlf the dimeter). r = D 2 = 8 2 = 4 m 3 Sustitute the vlue for r into the formul. A = π Clulte the result using π on your lultor. = Round to two deiml ples nd inlude the pproprite unit. 2 Clulte the re of eh irle. 15 mm 6.5 m 3 Clulte the re of eh irle. A m m d e f 19 m 18 mm 2

30 8E Are of irle 429 understnding AND FluenY 4 Clulte the re of irle with: rdius dimeter 7.5 m dimeter 15 mm d rdius 29 mm e rdius 0.8 m f dimeter 0.42 m 5 You n sutrt the re of one irle from nother to find the re of ring. Another nme for ring is n nnulus. Find the re for eh nnulus. 2 m 2 m 6 Use your understnding of re of irle to find the re of eh omposite shpe. 2 m e 4 m 13 mm 10 mm 9 mm 4 mm d f 11 m m 9 mm 15 mm 3 m 11 m 4 m 7 Anik wnts to mke irulr plemt. How muh mteril will she need if the plemt hs dimeter of 5? 8 A sprinkler sprys strem of wter 7.5 m long round in irle. Wht is the totl re overed y the sprinkler?

31 430 C HAPT E R 8 : M E A S U R E M E NT P r o l e m s o lv i n g n d r e s o n i n g 9 Wht is the lrgest pnke tht ould e ooked in this frying pn, if its se hs dimeter of 24 m? 10 Adrin wnts to ook pnke using the frying pn from question 9 nd ple it on this plte. PL E M Wht is the mximum re pnke ould hve if it hd to fit on the white setion of this plte nd the white setion hs rdius of? SA To mke pnke the size you found in prt, wht distne should e etween the edges of the pnke nd the side of the frying pn? Wht is the entire re of the plte if it hs dimeter of 30 m? d Wht is the re of the pinted setion of the plte? 11 A DVD hs dimeter of nd n inner (non-reordle) irle of dimeter 36 mm. There is 1 mm gp etween the edge of the DVD nd where the reording strts, nd there is n dditionl 3.5 mm ring tht nnot e reorded onto. Wht is the reordle re of the DVD? 1 mm ring 3.5 mm ring 36 mm 12 Wht is the totl re tht this utton overs if it hs dimeter of 20 mm nd eh hole hs rdius of 1.5 mm? 13 Design your own utton. Drw nd lel its dimensions, stting the re tht it will hve. 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:07 PM

32 8 E A r e o f irl e It is possile to find the rdius of irle if you know its re. A Explin why the formul to find the rdius of irle is r = π. Why might it e esier to find the length of the rdius if the re hs π in it? Support your nswer y finding the rdius of irle tht hs n re of: i 16 ii 16π. Wht mistkes might people mke when trying to find the rdius of irle using this formul? 15 Find the rdius of irle whih hs n re of: 9π m2 25π m2 100 m2 d Wht is the dimeter of irle tht hs n re of 49π m2? ple the dots onto domino set. You will need to reserh the numer nd type of tiles found in domino set. Assume n individul domino piee is 44 mm in length, 22 mm in width nd eh dot hs dimeter 4 mm. PL E Chllenge 17 Clulte the mount of pint required to 18 You n use knowledge of ngles to lulte the re M of setor. Look t the setor shown here. SA Wht is the ngle t the vertex of the setor? How mny degrees in irle? Use your nswers to prts nd to write the ngle for this setor s frtion of irle m d Wht is the rdius of this setor? e Wht would e the re of the irle if it ws omplete? f Use your nswers to prts nd e to find the re of the setor. g Explin why this works to lssmte. 19 Clulte the re of eh setor. 42 mm m 20 How ould you use your understnding of setors to find the res of the omposite shpes in question 6. 08_NOL_OM_SB8_29418_TXT_2pp.indd 431 Reflet Why is π importnt when working with irles? 20/01/14 10:07 PM

33 432 CHAPTER 8: MEASUREMENT 8F Surfe re Strt thinking! 1 How re figures A nd B relted? 2 Drw nets for figures C nd D. 3 How re the three nets similr nd how re they different? 4 A fe hs een lelled on eh ojet. Wht shpe is Fe 1 for the: retngulr prism? tringulr prism? squre-sed pyrmid? 5 Lel this Fe 1 on eh of your nets. 6 Lel the remining fes on eh of your nets. 7 Copy nd omplete this tle. The first fe for eh ojet hs een done for you. Retngulr prism Tringulr prism Squresed pyrmid Fe 1 Fe 2 Fe 3 Fe 4 Fe 5 Fe 6 Fe 1 Figure A Figure B Fe 1 Figure C 8 How would you use the informtion in the tle to lulte the totl surfe re of eh ojet? Complete 8F Disover tsk to lern more. Fe 1 Figure D Key ides The totl surfe re (TSA) of 3D ojet is the totl re of the outer surfe of the ojet. The TSA of prism is the sum of the res of eh fe of the prism. Using the net of n ojet mkes the fes esier to identify.

34 8F Surfe re 433 Exerise 8F Surfe re A <numers to ome> B <numers to ome> C <numers to ome> 1 Drw eh fe for these prisms. understnding AND FluenY e 15 mm 2 m 1 24 mm 25 mm 4 m d f 11 m 3 m m 13 m Exmple 8F-1 Clulting surfe re of retngulr prism Clulte the surfe re of this retngulr prism. 4 m Think 1 Identify the fes (three pirs of identil retngles). 2 Clulte the re of eh fe. 3 Add the res together nd inlude the pproprite unit. Write TSA = 4 m + 4 m m + 4 m = 40 m m m m m m 2 = 24 2

35 434 CHAPTER 8: MEASUREMENT 2 Clulte the surfe re of eh retngulr prism. understnding AND FluenY 2 m 12 mm 10 mm 4 m d 20 mm 3 m 2 m 4 m 2 m e f Exmple 8F-2 2 m Clulte the surfe re of this tringulr prism. Think 1 Identify the fes (two identil tringles nd three retngles). Clulting surfe re of tringulr prism Write 3 m 4 m TSA = m m Clulte the re of eh fe. = m m 2 3 Add the res together nd inlude the pproprite unit. = 152 m 2 3 Clulte the surfe re of eh tringulr prism. 4 m 9 m 20 m 3 m

36 8 F S urf e re u n de r s t n ding A N D F lu e n Y d 9 m m 2 m 11 m 1 24 m e f 13 m 4 m 9 m 4 A friend wnted to lulte the surfe re of this ue. She wrote her working s: TSA = E Wht is the surfe re of this ue? Is there shortut for finding the surfe re of ue? Explin it to lssmte. PL 5 Clulte the surfe re of ue tht hs side length of: 6m 12 mm d 20 m. M 4 m SA 6 Clulte the surfe re of eh pyrmid. Is there shortut? 9 m 13 m 8 mm 4 mm 11 m 8 mm 4 mm 7 Jess wnts to mke fitted sheet for the new mttress she ought. Assuming tht the mttress is retngulr prism, wht is the minimum mount of mteril tht she will need to uy in order to mke the fitted 1.75 m sheet? (Hint: the sheet hs to over the sides nd top of the mttress.) 08_NOL_OM_SB8_29418_TXT_2pp.indd m 2.1 m 20/01/14 10:07 PM

37 436 C HAPT E R 8 : M E A S U R E M E NT P r o l e m s o lv i n g n d r e s o n i n g 8 Thnh mkes ox for his mother in woodwork nd needs to over it in ontt inside nd out. How mny fes will Thnh need to over? Why is there more to over thn if it ws just losed ox Thnh deides to over it in lue ontt on the outer surfe nd white ontt on the inner surfe. 5 Clulte how muh lue ontt nd white ontt Thnh will need. Ignore the thikness of the wood. 1 4 d Wht is the totl surfe re tht Thnh will over in ontt? 9 Clulte the totl surfe re for eh open ox. 9 m 2 m 4 m 3 m E PL 10 Relulte the surfe re for eh ojet in question 9 if you only onsider the M outer surfe re. 11 If pool mesures 18 m y 11 m y 2 m, nd it osts $5 per squre metre to pint, SA how muh will it ost to pint the interior of the swimming pool? 12 Aron wnts to mke his own tent. He deides on simple tringulr prism design. If he wnts the tent to e 1 m tll, 2 m wide, 3 m long with slnted edge of 1.41 m, how muh mteril will he need? 13 Find the differene in rdord needed for smll lok of Tolerone ompred to lrge lok of Tolerone. (Hint: ssume the ends of eh Tolerone re equilterl tringles.) 30.5 m 5.1 m 14 A prism hs surfe re 5.9 m of 600 m2. Drw n exmple, lelling its dimensions, if this prism is: m m ue retngulr prism tringulr prism. 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:07 PM

38 8F Surfe re 437 Prolem solving nd resoning Chllenge 15 Why is there only one option for question 14 prt? 16 For eh speil qudrilterl: drw prism tht uses the qudrilterl s the se lel your drwing with pproprite dimensions; for exmple, height, width, length, et. find the totl surfe re for your prism d swp with lssmte, hek your nswers nd disuss ny differenes. 17 Find the minimum mount of lue pper nd red pper to wrp these gifts. Why would you need more thn the minimum mount of wrpping pper to over these gifts? 18 How muh ontt is needed to protet the over of this textook? Assume tht you over the spine nd oth sides of the front nd k of the ook. 19 How would you lulte the surfe re of tetrhedron? 20 m 2 20 m 3 20 Wht mistkes re ommon when finding the surfe re of n ojet? Consider the ue in question 4. Wht is its surfe re? Wht is the surfe re if two of these ues re stked, one on top of the other? Wht is the surfe re if there is stk of: i three ues? ii four ues? iii five ues? d Is there pttern? Desrie shortut to lulte the surfe re of stk of 17 ues. e Would this pttern still work with different sized ue? Investigte using ue with different dimensions. 22 How would you lulte the surfe re of ylinder? Reflet When ould knowledge of surfe re e useful?

39 438 CHAPTER 8: MEASUREMENT 8G Volume of prisms Strt thinking! 1 Wht type of prism is shown in figure A? 2 Wht is the shpe of its ross setion? Does it mtter where you slie the prism to otin the ross setion? This ross setion is lso known s the se of the prism. 3 m 3 If the se is the front fe, wht is the length of the dimension 4 m tht is t right ngles to this fe? This length is lled the height of the prism. 2 m Figure A 4 Does it mtter tht the height is not vertil length nd the se is not horizontl fe of the prism? The volume (V) of prism is found y multiplying the re of the Volume = re of se height se (A) y the height (H). V = A 5 Copy nd omplete the formul for volume of prism t right: or V = 6 Use the formul to show the volume in figure A is 24 m 3. 7 Does it mtter whih fe is the se? Show nother wy of working out the volume using the ottom fe s the se this time. Do you otin the sme nswer? 8 Wht type of prism is shown in figure B? (Hint: wht shpe is its ross setion?) 9 Wht is the re of its se? (Hint: this is the re of the ross setion.) 10 Wht is its height? 11 Use the formul to lulte its volume. 12 Could different fe of the prism e used s the se? Explin. 2 m 3 m 4 m Complete 8G Disover tsk to lern more. Figure B Key ides Volume is the mount of spe 3D ojet oupies. Common units of volume re ui millimetres (mm 3 ), ui entimetres (m 3 ) nd ui metres (m 3 ). A retngulr prism hs retngle for its se. A tringulr prism hs tringle for its se. Note We use H for height of prism so tht it is not onfused with h for height of tringle or prllelogrm. To find the volume (V ) of ny prism, use the formul V = AH, where A is the re of the se nd H is the height of the prism. The se nd height of prism must e perpendiulr (t right ngles) to eh other. The volume of other 3D ojets suh s ylinders n lso e found using the formul V = AH.

40 8G Volume of prisms 439 Exerise 8G Volume of prisms A <numers to ome> B <numers to ome> C <numers to ome> Exmple 8G-1 Clulting volume of retngulr prism Clulte the volume of this prism. 4 mm 7 mm understnding AND FluenY Think 3 mm Write 1 Write the formul. V = AH 2 Choose the front fe s the se. Identify the shpe (retngle) so pproprite re formul is used (A = lw). = l w H 3 Identify l, w nd H nd hek they re in the sme units. l = 3 mm, w = 4 mm, H = 7 mm 4 Sustitute the vlues into the formul nd lulte the result. Inlude the pproprite unit. 1 Clulte the volume of eh retngulr prism. 2 m 3 m 9 mm 3 mm 12 mm d V = = 84 mm 3 9 m 9 m 4 m e f 11 m 1 9 m 3 m 2 m

41 440 CHAPTER 8: MEASUREMENT Exmple 8G-2 Clulting volume of tringulr prism Clulte the volume of this prism. understnding AND FluenY Think 2 Clulte the volume of eh tringulr prism. 9 m 4 m 11 m d Write 1 Write the formul. V = AH 2 Choose front fe s se. Identify the shpe (tringle) so pproprite re formul is used (A = 1 2 h). 3 Identify, h nd H nd hek they re in the sme units. 4 Sustitute the vlues into the formul nd lulte the result. Inlude the pproprite unit. = 1 2 h H =, h =, H = V = = 150 mm m 29 m e f 1.9 m 9 m 21 m m 2.5 m

42 8 G V o lum e o f pris ms 441 u n de r s t n ding A N D F lu e n Y 3 Clulte the volume of eh prism. retngulr prism 2. long, 4. wide nd high tringulr prism 9 mm high, 3.5 mm wide nd with 12 mm se retngulr prism 7 m long, 2 m wide nd 11 m high d tringulr prism 5.5 mm high, 8 mm wide nd with 1.5 mm se e tringulr prism 1 high, wide nd with se f retngulr prism 11.5 mm long, 18 mm wide nd 7 mm high 4 Clulte the volume of eh ojet m 2 PL E 4 m 5. Exmple 8G-3 SA M 1 Clulting volume of n ojet Clulte the volume of this ojet. A = 2 Think Write 1 Write the formul. This ojet is not prism ut the sme formul n e used. V = AH 2 Identify the shpe of the se (irregulr). Its re is given. A = 2 3 Identify H nd hek its unit is entimetres. H = 4 Sustitute the vlues into the formul nd lulte the result. Inlude the pproprite unit. V = 8 10 = 80 m3 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:07 PM

43 442 C HAPT E R 8 : M E A S U R E M E NT u n de r s t n ding A N D F lu e n Y 5 Clulte the volume of eh ojet. The se re is shown. A = 2 A = 9 m2 A = 2 4 m d e 23.2 f 5.18 m2 16 mm2 9 m 3m 7 mm 6 Use the formul V = AH to lulte the volume of eh ojet. 4 m PL SA M 20 m E 1 P r o l e m s o lv i n g n d r e s o n i n g 7 A tissue ox is 21 m long, 11.4 m wide nd 12.3 m tll. Wht is its volume? If eh tissue lid flt in the ox is 1 mm high, how mny tissues would it fit? 2.5 m 2.0 m 8 Explin why the volume of ny irregulr prism n e found y using the formul V = AH. 1.3 m 9 A skip hs the dimensions shown. Clulte its volume in ui metres. 1.5 m 10 A new kind of insultion is to e fitted in the roof of house. It is mesured in ui metres rther thn squre metres euse it fills the entire roof (normlly insultion is just thik lyer). The roof is in the form of tringulr prism. If the roof is 2 m high, 9 m wide nd 14 m long, lulte: how muh insultion is needed (in ui metres) how muh it will ost if the insultion osts $30 per ui metre. 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:08 PM

44 443 8 G V o lum e o f pris ms P r o l e m s o lv i n g n d r e s o n i n g 11 Andrew likes to mke geometri sulptures. If his mterils ost $2 per ui metre, lulte how muh it osts him to produe eh sulpture. 1m 2.5 m 1.8 m 3m 1.8 m 2m 1m 4m 1.8 m 1.5 m 1m 2m 2.5 m 2m 12 Mel deides she wnts one of the sulptures from question 11. She n t deide whih one, ut it needs to e s light s possile. Use the informtion t right to reommend whih sulpture Mel should uy. 1m 3m 4m 1m Sulpture : weighs 3 kg per ui metre Sulpture : weighs 2 kg per ui metre Sulpture : weighs 2.5 kg per ui metre 13 A retngulr prism hs squre se with sides tht mesure. If the prism hs E volume of 100 m3, wht is its length? PL 14 A tringulr prism must e 20 m long nd hve volume of 400 m3. Wht re must the tringle fe e? M If the tringle fe must e extly high, wht is the se mesurement? SA If the tringle fe must e less thn high, wht re three possile sets of dimensions (se nd height) of the tringle fe? 15 If you were to fit yourself into rdord ox, wht would e: the smllest possile set of dimensions for this ox? the smllest possile volume it ould hve? 16 A retngulr prism hs volume of 500 m3. List three possile sets of dimensions. 17 A tringulr prism hs volume of 500 m3. List three possile sets of dimensions. 2 Drw these three exmples. Try to use different shped tringles for eh one. 30 m when finding the volume of prism? Chllenge 08_NOL_OM_SB8_29418_TXT_2pp.indd m Find the volume of onrete needed to 2 mke these two loks t right m3. Explin where Juli went wrong nd help her find the orret nswer Wht ommon mistkes do people mke 20 Juli found the nswer to question 19 to e 2 30 m Reflet How is the volume of prism relted to its se? 20/01/14 10:08 PM

45 444 CHAPTER 8: MEASUREMENT 8H Are nd volume onversions Strt thinking! 1 Wht re the dimensions of figure A? How mny squre entimetres in figure A? 2 Wht re the dimensions of figure B? How mny squre millimetres in figure B? Is this re the sme or different from figure A? d Explin why, even though there re 10 mm in 1 m, there re 100 mm 2 in 1 m 2. 3 Drw nother squre nd lel its sides with 1 m. Use this squre to explin why, even though there re 100 m in 1 m, there re m 2 in 1 m 2. 4 How mny ui entimetres in figure C? Copy nd relel figure C in millimetres. Explin why, even though there re 10 mm in 1 m, there re 1000 mm 3 in 1 m 3. Complete 8H Disover tsk to lern more. Key ides 1 m Figure A 1 m Figure C 10 mm Figure B When onverting etween units of re, you multiply/divide y the squre of the onversion ftor For exmple, s 1 m = 10 mm: 1 m 1 m = 1 m 2 km 2 m 2 m 2 mm 2 10 mm 10 mm = 100 mm so 1 m 2 = 100 mm 2 or 10 2 mm 2 When onverting etween units of volume, you multiply/divide y the ue of the onversion ftor For exmple, s 1 m = 100 m: 1 m 1 m 1 m = 1 m 3 km 3 m 3 m 3 mm m 100 m 100 m = m so 1 m 3 = m 3 or m 3 Cpity is the mount of fluid tht ontiner n hold. Common units for pity re millilitres (ml), litres (L), kilolitres (kl) nd meglitres (ML).

46 8H Are nd volume onversions 445 Exerise 8H Are nd volume onversions A <numers to ome> B <numers to ome> C <numers to ome> understnding AND FluenY 1 Copy nd omplete: To onvert etween entimetres nd millimetres the onversion ftor is 10. To onvert etween m 2 nd mm 2 the onversion ftor is 10 2 = = To onvert etween m 3 nd mm 3 the onversion ftor is 10 3 = = To onvert etween metres nd entimetres the onversion ftor is 100. To onvert etween m 2 nd m 2 the onversion ftor is = = To onvert etween m 3 nd m 3 the onversion ftor is 100 = = To onvert etween kilometres nd metres the onversion ftor is To onvert etween km 2 nd m 2 the onversion ftor is 1000 = = To onvert etween km 3 nd m 3 the onversion ftor is 1000 = = 2 Copy nd omplete these onversion sttements. The first one hs een done for you. 2 into mm 2 = ( ) mm 2 = (6 100) mm 2 = 600 mm 2 4 m 2 into m 2 = ( ) m 2 = (4 ) m 2 = m mm 2 into m 2 = ( ) m 2 = (900 ) m 2 = m 2 d 3 into mm 3 = ( ) mm 3 = (5 1000) mm 3 = mm 3 e 7 m 3 into m 3 = ( ) m 3 = (7 ) m 3 = m 3 f 2000 mm 3 into m 3 = ( ) m 3 = (2000 ) m 3 = m 3 Exmple 8H-1 Converting to smller re unit Convert 3. 2 into mm 2. Think 1 1 m = 10 mm so 1 m 2 = 10 2 mm 2. Multiply y onversion ftor (10 2 ) to onvert to smller unit. Write 3. 2 = ( ) mm 2 2 Clulte (10 10 = 100) = ( ) mm 2 3 Complete the multiplition. = 380 mm 2

47 446 CHAPTER 8: MEASUREMENT 3 Convert eh mesurement. 3 m 2 into m 2 2 into mm m 2 into m 2 d into mm 2 e 1.6 km 2 into m 2 f 0.75 km 2 into m 2 Exmple 8H-2 Converting to lrger re unit Convert m 2 into m 2. Think 1 1 m = 100 m so 1 m 2 = m 2. Divide y onversion ftor (100 2 ) to onvert to lrger unit. Write m 2 = ( ) m 2 2 Clulte ( = ) = ( ) m 2 3 Complete the division. = 10 m 2 Exmple 8H-3 Convert 2 m 3 into m 3. Think 4 Convert eh mesurement m 2 into m mm 2 into m mm 2 into m 2 d m 2 into km 2 e 67 2 into m 2 f m 2 into km 2 Converting to smller volume unit 1 1 m = 100 m so 1 m 3 = m 3. Multiply y onversion ftor (100 3 ) to onvert to smller unit. Write 2 m 3 = ( ) m 3 2 Clulte ( = ) = ( ) m 3 3 Complete the multiplition. = m 3 5 Convert eh mesurement. 9 m 3 into mm 3 18 m 3 into m km 3 into m 3 d 4.56 m 3 into m 3 e 0.15 km 3 into m 3 f 6.1 m 3 into mm 3

48 8H Are nd volume onversions 447 Exmple 8H-4 Converting to lrger volume unit Convert 216 mm 3 into m 3. Think 1 1 m = 10 mm so 1 m 3 = 10 3 mm 3. Divide y onversion ftor (10 3 ) to onvert to lrger unit. Write 216 mm 3 = ( ) m 3 2 Clulte ( = 1000) = ( ) m 3 3 Complete the division. = understnding AND FluenY Prolem solving nd resoning 6 Convert eh mesurement mm 3 into m m 3 into km m 3 into m 3 d m 3 into m 3 e 28 mm 3 into m 3 f m 3 into km 3 7 Convert eh mesurement into mm m 2 into m m 2 into km 2 d mm 3 into m 3 e 18 2 into m 2 f km 3 into m 3 g 45 2 into mm 2 h 75 m 3 into km 3 i 1 m 3 into m 3 j 1.32 km 2 into m 2 k m 3 into m 3 l 16 mm 2 into m 2 8 Ahmed needed to onvert 11 m 2 into mm 2. He sid tht euse there re 10 mm in 1 m, then there re 110 mm 2 in 11 m 2. Explin where he went wrong nd provide the orret nswer. 9 Jess ompleted worksheet of onversions, shown elow. i m 2 = ( ) m 2 ii 200 m 2 = ( ) mm 2 = 400 m 2 = mm 2 iii 7 m 3 = ( ) m 3 iv 3000 mm 3 = ( ) m 3 = 2100 m 3 = 30 m 3 Mrk her work nd give her sore out of 4. Provide suggestion for why she mde eh mistke. Provide the orret working for the prolems tht she got inorret.

49 448 CHAPTER 8: MEASUREMENT Prolem solving nd resoning 10 A ompny produes pinted wooden toy oxes. Clulte the surfe re of the outside of the ox (ll six fes re to e pinted). If eh ox needs two ots of pint nd one tin of pint overs 15 m 2, how mny toy oxes n e pinted with one tin? 11 The ompny would like to know the volume 60 m of the ox from question 10 in ui metres, 10 ut the dimensions re in ui entimetres. Clulte the volume in ui entimetres. Convert this to ui metres. Another wy to do this is to first onvert ll the dimensions into metres nd then find the volume. Show how doing the onversion first gives you the sme nswer to prt. 12 Aron hrvests whet on his frm nd gets yield of out 3.2 tonnes per hetre. A hetre is m 2. Write this in: i kilogrms per hetre ii kilogrms per squre metre iii grms per squre metre iv grms per squre entimetre Whih unit do you think mkes the most sense to use? Explin. If Aron is pid 35 ents per kilogrm of whet, find how muh money he will ern if he hrvests 3000 hetres of whet. 13 Rememer tht hetre is m 2. How mny hetres re in 1 km 2? There re mny other units of re tht re still used tody tht re not metri, suh s the re. A hetre is roughly equl to 2.5 res. Approximtely how mny squre metres re in n re? Investigte the ext onversion ftor etween n re nd metri units suh s the squre metre nd hetre. d Investigte other units of re; for exmple, squre mile, squre yrd, nd explin how they relte to the metri units of re suh s the squre metre. 14 Look t this onversion hrt for pity. How mny millilitres in litre? Convert: i 7.5 L into ml ii 450 ml into L iii 900 kl into ML ML kl L ml iv 3.75 kl into L

50 P r o l e m s o lv i n g n d r e s o n i n g A ontiner of volume 1 m3 holds 1 ml of liquid. Similrly, 1 m3 holds 1 kl of liquid. Find the volume of the ontiner (in m3) tht would hold: i 15 ml ii 1 L iii 5 L Find the volume of the ontiner (in i 4 kl ii 13 kl iv 1 kl m3) iii 200 L tht would hold: iv 745 L Find the pity held in ontiner with volume of: i 23 ii 5000 m3 iii 9 m3 iv 0.4 m3 16 Clulte the volume of this perfume ottle in ui entimetres. Clulte its pity in litres. 17 A retngulr th tu mesures 1.5 m long y 30 m wide, with depth of 4. How mny litres of wter ould it hold? 18 A r engine is sid to e This mens tht the ylinders E in the engine hve volume of 1200 m3. How mny litres of petrol n the ylinders hold? 11 m 19 A retngle hs n re of 4800 mm2. Give three possile sets of dimensions in entimetres. PL 2. 3 m 20 A retngulr ox hs volume of m3. millimetres M Give possile set of dimensions in: entimetres metres. SA 4 8 H A r e n d v ol ume o nv e rs io ns 21 Write your own open-ended question out re nd volume nd swp with lssmte. Be reful of onversion errors. Disuss ny differene in nswers. Chllenge 22 Melourne s wter storges hve pity of ML. Write this in: i kilolitres ii litres iii ui metres iv ui kilometres. Whih expression do you think hs the most mening for people? Explin. If the wter storges re only 47% full, how mny kilolitres is this? d Melourne Wter estimtes tht eh person uses 300 L of wter per dy. i How mny litres of wter is this per yer? ii How mny kilolitres of wter is this per yer? e If there re out 4 million people living in Melourne, how mny kl of wter is this yer? f If the storges re 47% full, use your nswer from prt e to find how quikly our storges would run out if there ws no rin. Answer to the nerest yer. g Wht other uses of wter re there tht would tke wter from storges? 08_NOL_OM_SB8_29418_TXT_2pp.indd 449 Reflet Wht ommon mistkes might people mke when onverting re or volume units? 20/01/14 10:08 PM

51 450 CHAPTER 8: MEASUREMENT CHAPTER REVIEW Summrise 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. length perimeter estimte irumferene dimeter rdius re setor Multiple-hoie 8A 8B 8B totl surfe re volume pity width retngle tringle se height 1 Wht is 7.84 m equl to? A km B 7840 mm C 78.4 m D km 2 A irle hs dimeter of. Wht is its rdius? A B 6π m C 36π m D 3 m 3 Wht is the totl perimeter of this shpe? 15 mm dimensions perpendiulr omposite shpes qudrilterls prllelogrm rhomus kite trpezium 8D 8E 8F nnulus net retngulr prism tringulr prism squre-sed prism ue pyrmids onversion ftors 5 A rhomus hs digonls of lengths nd. Wht is its re? A 30 m 2 B 11 m 2 C 5 2 D Wht is the re of the shpe in question 3? A mm 2 B 754 mm 2 C mm 2 D 240 mm 2 7 Wht is the totl surfe re of ue with side length 5 mm? A 150 mm 2 B 125 mm 2 C 25 mm 2 D 15 mm 2 8C A 31 mm C 63.1 mm 16 mm B 47 mm D 79.1 mm 4 A tringle hs se length of nd n re of 20 m 2. Wht is its height? A 2 m B 4 m C D 200 m 8G 8H 8 Wht is the volume of n irregulr prism with se re 2. 2 nd length 10 mm? A 250 mm 3 B 25 mm 3 C 2. 3 D Wht is 3 m 3 equl to? A 30 mm 3 B 300 mm 3 C 300 m 3 D 3000 mm 3

52 451 8 C h pt e r Re v ie w Short nswer 8A 1 Clulte the perimeter of eh shpe. 8E 5 Clulte the re of eh irle in question 2. 8F 6 Clulte the totl surfe re of eh 2 m 25 mm 3.9 m 1.9 m ojet. 3.2 m 35 mm 3 m 56 mm 2 mm 4 m 10 mm d 7 mm 4 mm 10 mm 13 mm 3 mm 7 mm 5 mm 2 Clulte the irumferene of eh 8 mm d 19 m 13 m 2 m 1 8G 1 9 Convert eh mesurement. 5 m2 into m2 11 m d m 9 m m m2 rik. 18 mm 3 m 11 8 Clulte the volume of ly in this 4 Clulte the re of eh shpe. 08_NOL_OM_SB8_29418_TXT_2pp.indd 451 A= 8G 20 m 8D d 13 m 1 9 m 1 4 m 3 Clulte the re of eh shpe. 9 m SA 8C 6 mm M 3 mm 7 Clulte the volume of eh ojet. PL irle. 8G 3 mm E 8B 4 mm 5 mm 6 mm 50 mm2 into m2 20 m 3.3 into mm3 d 400 m3 into m3 e 7.5 L into ml f 845 kl into L. 20/01/14 10:08 PM

53 452 CHAPTER 8: MEASUREMENT NAPLAN-style prtie 1 Jun is uilding model eroplne. He needs plnk of wood tht is 1150 mm long. If he uys plnk of wood 1.5 m long, how mny entimetres will he hve to ut off? The totl length of ross-ountry Strt 500 m ourse is 5 km. Wht is the length 1.7 km of the lst leg? 700 m 2.1 km 4.3 km 9.3 km 430 m 3 A vegetle grden is in the shpe of retngle. The totl length of fening is 3.4 m nd the grden is 1 m long. How wide is the grden? 4 A irulr sushi hnd roll mesures 4. ross its dimeter. Wht length of seweed wrps round the irumferene? 5 A irulr ht mesures 6 round the rim. How wide is it ross its dimeter? m km m 10.3 Questions 6 nd 7 refer to r wheel with rdius of 20 m. 6 Wht is the length of one revolution of the wheel? 20 m m m 7 By how muh would this rdius need to inrese if the wheel hd to over 1.5 m in one revolution? m Questions 8 nd 9 refer to the following informtion nd figure. 24 m A pool hs onrete 20 m pvers round its 14 m edges, s shown in the figure. 8 Wht is the re of the swimming pool? 9 Wht is the re overed y the onrete pvers? 336 m m m 2 10 Derek designed 30 m this flg for his sports tem. Wht is the 1 re of the lue setion? m m 2 11 A tringle hs n re of 40 m 2. If it is high, how long is its se? 12 Brittny hs kite tht is 7 long nd 40 m wide. Wht is the re of this kite? 13 Wht is the re of this shpe? 14 A Frisee hs dimeter of 2. Wht is its re? 10 m 20 m 15 A pir of errings is to e mde ording to the lueprints shown. How muh metl is needed to mke the pir? m m m m 1.6 m 5.9 m 5.9 m 2.4 m 3.1 m 5.9 m

54 8 Chpter Review Wht is the dimeter of the lrgest plte tht ould e mde with 200 m 2 of porelin (to the nerest entimetre)? 4 m 1 64 m 19 Wht is the differene in volume etween these ontiners? 263 m m m 3 Questions refer to the following informtion. Tennis lls re pled into ontiner tht is retngulr prism with the dimensions y y 21 m. 17 How muh rdord is required to mke this ontiner? 18 The tennis lls re insted pled inside ontiner in the shpe of n equilterl tringulr prism, s shown. How muh rdord is required to mke this ontiner? 44 m m m m 2 21 m 13 m Anlysis Questions refer to the following informtion. Ktie hs vse in the shpe of retngulr prism. Its se mesures y nd it is 20 m high. 20 Wht is the volume of the vse? 21 Wht mount of wter will fit inside the vse? 5 L 50 L 500 L 25 ml 22 How mny vses would e required to hold 1 m 3 of wter? A sheet of glss hs n re of 3500 mm 2. Wht is this mesurement in m 2? A friend wnts to hve ll pit t their irthdy prty. They hve set side retngulr spe in their kyrd tht is 3 m long nd 2 m wide. Wht re would this over? How muh fening would e needed to go round the ll pit? Your friend n only uy the wood tht they need for the fening in plnks tht re 50 m wide. The wlls of the pit need to stnd 1 m tll. Wht length of wood will e needed for the fene? d Your friend wnts to pint the wooden fening inside nd out with two ots of pint. Wht is the totl surfe re tht they will need to over? They lso wnt to pint dimond on the front fene of the ll pit. From point to point (tht is, its digonls) it mesures 5 vertilly nd 30 m horizontlly. e Wht is the re of this dimond? Around the dimond will e six irles representing the oloured lls in the ll pit. Eh ll hs dimeter of 6.. f Wht is the irumferene of eh irle? g Find the totl re tht needs to e pinted to represent the six lls. h Wht will e the volume ontined within the pit? i If eh ll to go into the pit hs volume of 150 m 3, how mny ould theoretilly fit into the pit? Be reful with unit onversions! j Your friend finds tht only out lls fit into the pit. Wht explntion n you offer for this differene?

55 454 C HAPT E R 8 : M E A S U R E M E NT CONNECT Plnning house SA M PL E Plnning nd onstruting ny uilding requires good understnding of mesurements nd mthemtis. Wht mesurements re needed when plnning to uild house? Your tsk For your investigtion onsider the design nd ost of onstruting simple house, suh s uy house, dog kennel, doll house or even irdhouse. You will need to omplete these steps. Reord the type nd purpose of house. Prepre digrms showing the shpe, size nd dimensions of the house Also prepre digrms to show the shpe, size nd dimensions of house fetures (for exmple, doors, windows, lonies). Mke lists of the type nd ost of uilding mterils (for exmple, wood, plsti, glss, tiles) Work out the ost of pinting the house, inside nd out, nd other deortion. 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:08 PM

56 PL SA M Complete the 8 CONNECT worksheet to show ll your working nd nswers to this tsk. You my like to present your findings s report. Your report ould e in the form of: tlogue lueprints rohure other (hek with your teher). 455 E 8 o n n e t 08_NOL_OM_SB8_29418_TXT_2pp.indd /01/14 10:08 PM