Area and Volume. Introduction

Transcription

1 CHAPTER 3 Are nd Volume Introduction Mn needs mesurement for mny tsks. Erly records indicte tht mn used ody prts such s his hnd nd forerm nd his nturl surroundings s mesuring instruments. Lter, the imperil system ws introduced wherey the units of length were inches, feet, yrds nd miles. Tody we use the metric system, which ws introduced in the 18th century. The metric system uses the sic unit of the metre. This is very convenient system ecuse ll the units re sed on multiples of 10 nd, hence, it is esy for conversion purposes. All of these were studied in some detil in Yer 7. Tody mny people re interested in do-it-yourself projects. Mny hrdwre stores offer free clsses for people to lern how to tile or crpet their floors or pint the wlls of their house, s well s mny other interesting tsks. You my wish to help your dd complete such project. In this chpter, we will lern such things s how to find the re required to tile or crpet floor, or how to clculte the mount of pint required to pint the wlls of house, or the mount of wter required to fill swimming pool. In order to do this, we will need to understnd few concepts such s: re the mount of surfce covering closed shpe volume the mount of mteril of which solid is mde cpcity the mount of spce inside continer tht cn e filled with solid, liquid or gs solid, which occupies spce, hs definite shpe nd cn e seen nd touched. More informtion on these concepts will e delt with in detil in this chpter.

2 76 oxford mthemtics for victori A R E A A N D V O L U M E W A R M U P 1 Drw ech shpe nd highlight the perimeter. 4 Clculte the re of ech shded section. c d c Clculte the perimeter of ech digrm. c 4 cm d 8.5 cm 4.3 cm d 35 mm 4 mm 3 Find the re of the following. 5 Convert the following to the units given in rckets. 3 mm 9 cm 10 cm (mm) 5.5 m (cm) c 1.8 km (m) d 50 mm (cm) 48 mm 15 cm e 40 cm (m) f 500 m (km) c d g 5 cm (mm ) h.7 cm (mm ) i 1 m (cm ) j 11.8 m (cm ) 17 cm k 5 h (m ) l 3 km (h) m 600 mm (cm ) n cm (m ) o m (h) p 750 h (km ) e f

3 3A Specil types of qudrilterls A qudrilterl is figure tht hs four sides. The lengths of the sides nd the size of the ngles in qudrilterl determine its nme. This flow chrt descries the specil types of qudrilterls. re nd volume 77 One pir of opposite sides re prllel Two opposite sides re equl. Two djcent sides re equl. All four sides re equl. All four sides re equl nd prllel. Two opposite sides re equl nd prllel. Fetures of these specil qudrilterls Shpe Nme Properties Rectngle Squre Prllelogrm Two opposite sides re equl. Digonls re equl. Digonls isect (cut eqully) ech other. The ngle t ech vertex is 90. All sides re equl. Digonls re equl. Digonls isect ech other t right ngles (90 ). The ngle t ech vertex is 90. The digonls isect ech ngle t the vertex. Two opposite sides re equl nd prllel. Digonls re not equl. Digonls isect ech other. Opposite ngles re equl. Adjcent ngles re supplementry (dd up to 180 ).

4 78 oxford mthemtics for victori Shpe Nme Properties Rhomus Kite Trpezium Isosceles trpezium All four sides re equl nd prllel. Opposite ngles re equl. Digonls isect ech other t right ngles (90 ). Digonls isect ech ngle t the vertex. Adjcent ngles re supplementry. Two pirs of djcent (next to) sides re equl. One pir of opposite ngles re equl. Digonls isect ech other t right ngles (90 ). One digonl isects ech ngle t the vertex. It is symmetricl out one digonl. One pir of opposite sides re prllel. Two non-prllel sides re not equl. Digonls re not equl. Digonls do not isect ech other. One pir of opposite sides re prllel. Two non-prllel sides re equl. Digonls re not equl. Digonls do not isect ech other. A digonl is the line joining two opposite non-consecutive vertices of polygon. A polygon is closed figure mde up of mny ( poly ) stright lines. You will lern more out this lter. Are conversions In Yer 7, we lernt how to convert units in re. Recll the following: km h m cm mm Rememer To convert from lrger unit to smller unit, you multiply. To convert from smller unit to lrger unit, you divide. Ares of composite shpes Composite shpes re mde up of more thn one regulr shpe. To find the re of composite shpe, find the re of ech regulr shpe nd dd or sutrct them.

5 re nd volume 79 I N V E S T I G A T I O N Are of specil qudrilterls 1 Find the re of ech tringle. c 1 cm 10 cm Are of tringle 5 cm 8 cm 5 cm Are of tringle 5 cm Hence: Are of the prllelogrm 5 3 units height se e The re of prllelogrm cn e lso otined y the following method. Investigtion continued next pge 1 cm Are of tringle 5 cm Find the re of the shded region in ech prllelogrm: Are of tringle 5 cm 8 cm length = se width = height Are of prllelogrm 5 re of rectngle 5 length 3 width 5 se 3 height 4 Find the re of the shded region. 10 cm 7 cm Are of tringle 5 cm c Put oth shded regions together. The re of the prllelogrm 5 1 cm d This re cn e lso otined s follows. Are of prllelogrm 5 1 (14 3 ) 1 1 (14 3 ) cm 5 (14 3 ) cm (since ) 5 cm (14 cm is the se nd 7 cm is the height of the prllelogrm.) 1 cm Are of tringle 5 cm 10 cm 7 cm 1 cm Are of tringle 5 cm

6 80 oxford mthemtics for victori Investigtion continued c Put oth shded regions together. Are of the trpezium 5 1 cm d Are of the trpezium cn e lso otined s follows: Are of the trpezium cm ( 1 10) cm (1 cm nd 10 cm re the prllel sides nd 7 cm is the height of the trpezium.) Hence: Are of trpezium height 3 ( 1 ) Where nd re the prllel sides nd h is the height (perpendiculr distnce etween them). 5 Find the re of the shded region. Are of tringle 5 cm Are of tringle 5 cm c Put oth shded regions together. Are of the rhomus 5 1 cm d Are of the rhomus cn e lso otined s follows. Are of the rhomus cm ( 1 ) cm cm h (1 cm nd 6 cm re the digonls of the rhomus.) Hence: Are of rhomus units product of Where d 1 nd d re the digonls of the rhomus. 8 Find the re of the shded region. Are of tringle 5 cm Are of tringle 5 cm Put oth shded regions together. c Are of the kite 5 1 cm d Are of the kite cn e lso otined s follows: Are of the kite cm ( 1 ) cm cm (14 cm nd 8 cm re the digonls of the kite.) Hence: Are of kite units product of Where D 1 nd D re the digonls of the kite.

7 re nd volume 81 From the investigtion, the re of ech specil type of qudrilterl cn e summrised s given in the tle. Nme Shpe Formul for re 1 Rectngle Are of rectngle 5 l 3 w w where l is the length nd w is the l width of the rectngle. Squre Are of squre 5 l l l where l is the side of squre. 3 Prllelogrm Are of prllelogrm 5 3 h h where is the se nd h is the height of the prllelogrm. 4 Rhomus Are of rhomus d 3 d 1 where d 1 nd d re the digonls of the rhomus. 5 Kite Are of kite D 3 D 1 where D 1 nd D re the digonls of the kite. 6 Trpezium Are of trpezium h( 1 ) h where nd re the prllel sides nd h is the height of the trpezium. Other types of qudrilterls tht you should know exist re given here. A concve qudrilterl is one in which one or more digonls re outside the qudrilterl. A convex qudrilterl is one in which ll of the digonls re inside the qudrilterl. All the specil types illustrted ove re convex. A cyclic qudrilterl hs ll the vertices on the circumference of circle.

8 8 oxford mthemtics for victori E X A M P L E S Exmple A1 Convert the following m to cm m to h c cm to m Lrger 3 $ smller Smller 4 $ lrger Use the units of conversion m cm cm m h 5 60 h c cm m 5 35 m Exmple A Find the re of: rectngle with length 1.5 cm nd width 5.6 cm squre with sides 4.8 cm c prllelogrm with se 11 cm nd height 7 cm. Are of rectngle 5 l 3 w Are of squre 5 l 3 l c Are of prllelogrm 5 3 h Use the formule. Are of rectngle 5 l 3 w (length 3 width) cm (Use your clcultor.) 5 70 cm Are of squre 5 l 3 l (side 3 side) cm (Use your clcultor.) cm c Are of prllelogrm 5 3 h (se 3 height) cm 5 77 cm

9 re nd volume 83 E X A M P L E S Exmple A3 Find the re of rhomus with digonls d cm nd d 5 13 cm. Find the re of kite with digonls D cm nd D 5 6 cm. Exmples continued next pge Are of rhomus 5 1_ 3 product of digonls Are of kite 5 1_ 3 product of digonls Are of rhomus 5 1_ 3 product of digonls 5 1_ 3 d 3 d 1 5 1_ cm Are of kite 5 1_ cm 3 product of digonls 5 1_ 3 D 3 D 1 5 1_ cm 5 33 cm Exmple A4 Find the re of trpezium with prllel sides 11 cm nd 8.5 cm, nd with perpendiculr distnce etween them (height) of 6.3 cm. 8.5 cm 6.3 cm 11 cm Are of trpezium 5 1_ 3 h( 1 ), where nd re the prllel sides nd h is the height. Use the formul. Are of trpezium 5 1_ 3 h( 1 ) 5 1_ 3 6.3( ) cm 5 1_ cm (Use your clcultor.) cm

10 84 oxford mthemtics for victori Exmples continued Exmple A5 The re of prllelogrm is 77.5 cm. If the se is 5.5 cm, find its height. Are of prllelogrm 5 3 h, where 5 se nd h 5 height. Hence: h 5 A Use the formul. Height of prllelogrm 5 re 4 se (Use your clcultor.) cm (correct to 1 deciml plce) Exmple A6 Find the re of this shpe. All mesurements re in millimetres. 7 mm 9 mm 9 mm 13 mm Are of trpezium 5 1_ 3 h( 1 ) Are of prllelogrm 5 3 h Use the formule nd dd the res ecuse the shpe is mde up of trpezium nd prllelogrm. Are of trpezium 5 1_ 3 h( 1 ) 5 1_ 5 1_ 3 7 (9113) mm mm 5 77 mm Are of prllelogrm 5 se 3 height mm mm Hence, re of shpe 5 ( ) mm mm

11 re nd volume 85 Exmple A7 A ckyrd is in the shpe of trpezium. There is squre flowered nd tringulr vegetle grden. Aron wnts to sow some lwn seeds round the grden. Find the re of the lwn. (All mesurements re in metres.) Are of squre 5 l 3 l Are of tringle 5 1_ 3 3 h Are of trpezium 5 1_ 3 h( 1 ) Use the formule to find the res of the squre nd the tringle, nd dd them. Then find the re of the trpezium nd sutrct the res of the flowered nd the grden to give the re of the lwn. Are of squre 5 l 3 l m (Use your clcultor.) m Are of tringle 5 1_ 3 3 h 5 1_ m m Are of squre nd tringle 5 ( ) m Are of trpezium 5 1_ 5 1_ 5 1_ 5 1 m 3 h( 1 ) 3 7.3( ) m m m Are of lwn 5 (91.5 1) m m Exercise 3A 1 Convert the following to the units given in rckets cm (m ) 5 h (m ) c h (m ) d 3 m (mm ) e 13.7 m (cm ) f m (cm ) g 30 cm (mm ) h 3 m (cm ) i m (mm ) j 0.09 cm (mm ) k m (h) l 3 km (h) m km (h) n 000 h (km ) o m (km ) p h (km )

12 86 oxford mthemtics for victori Giving your nswer correct to deciml plces, find the re of ech of the following. se 5 7 mm, height mm se 5 9 mm, height mm h c l cm, w 5 6 cm d l mm, w mm w w l l e side cm f side mm g 5 11 mm, h 5 1 mm h mm, h cm h i cm, h 5 3.3cm j d mm, d cm d 1 d k d m, d 5 30 cm l d mm, d 5 1. cm d 1 d d 1 d

13 re nd volume 87 m d mm, d mm n d cm, d 5 16 m d d 1 d 1 d o d cm, d 5 17 cm p l mm, w mm d d 1 l w 3 Find the re of ech trpezium. Give your nswers to two deciml plces. 6 cm 7.8 cm c 7 cm 4.3 cm 5 cm 9 cm 11 cm 3. cm 13 cm d 4 cm e f 11.3 cm 6 cm 3.4 cm 11 cm 10.3 cm 4 For ech of the following, give your nswer to the nerest integer. Find if the re is 4.5 m Find h if the re is 9.6 m nd the height is 150 cm. nd the se is 30 cm. c Find if the re is 680 cm, the d Find the long digonl if the re is height is 17 cm nd the se is 46 cm. 380 mm nd the short digonl is 19 mm. e Find h if re mm, f Find h if re 5 70 cm nd se 5 3 cm mm nd 5 40 mm.

14 88 oxford mthemtics for victori 5 Find the re of composite shpe. Give your nswer correct to deciml plces. c d e f 6 Giving your nswer correct to deciml plces, find the re of: i the shded section ii the unshded section. c d 7 Pving tiles mesuring 375 mm y 375 mm re used to pve the re shown elow. Find the re to e pved. Find the re of one tile. c How mny tiles re required to pve this re? d Find the totl cost of pving the re if the tiles cost $3.95 per squre metre, other mterils such s snd nd crushed rock cost $68 with 10% GST nd lour costs $30 per squre metre.

15 re nd volume 89 8 For her new home, Jcqueline selects cermic floor tiles mesuring 450 mm y 450 mm. How mny tiles re required to tile three rooms with floors mesuring 6 m y 4.5 m? If the tiles chosen cost $4.50 per squre metre plus 10% GST nd lour costs $5 per squre metre, find the totl cost of lying the tiles. 9 A rectngulr lwn is surrounded y concrete pth, which is 1.5 m wide. The lwn mesures m y 8 m. Drw digrm nd find the re of the lwn. Find the re of the concrete pth. 10 A pth tht is 0.8 m wide is uilt round two flowereds, s shown in the figure. The two flowereds re identicl in shpe nd size. Find: the width of flowered the length of flowered c the re of flowered d the re of the pth. 3B Circles A circle is shpe enclosed y curved line. This curved line is its oundry. In Yer 7, we lernt tht the oundry of closed shpe is clled its perimeter nd the perimeter of circle is clled the circumference. We lso hve lernt the formul for clculting the circumference of circle. Circumference of circle 5 pr or Circumference of circle 5 pd, since D 5 r It would e interesting to know the different fetures of circle. Fetures of circle No. Nme of feture Digrm Definition 1 Centre A point in the middle of the circle equidistnt from ll points on the circumference. The ngle t the centre of the circle is 360. Circumference The oundry line enclosing the circle. The perimeter of the circle.

16 90 oxford mthemtics for victori No. Nme of feture Digrm Definition 3 Rdius The line joining the centre of the circle to ny point on the circumference of the circle. 4 Dimeter A line joining two points on the circumference nd pssing through the centre. 5 Chord A line joining two points on the circumference ut not pssing through the centre. 6 Tngent A line touching the circle t one point nd forming n ngle of 90 with the rdius. 7 Mjor rc Minor rc Ech is frction of the whole circumference. 8 Mjor sector Minor sector A sector is frction of the surfce of the whole circle. It is ounded y two rdii nd n rc. The two rdii mke n ngle t the centre of the circle. If θ is the ngle t the centre, then frction of the circle 5 θ Semicircle Hlf circle ounded y the dimeter nd n rc. 10 Qudrnt One qurter of circle, ounded y two rdii nd n rc. It is sector with the ngle t the centre t Concentric circles Two or more circles hving the sme centre.

17 re nd volume 91 No. Nme of feture Digrm Definition 1 Mjor segment (lrger section) A segment is prt of circle ounded y chord nd n rc. Minor segment (smller section) The length of n rc is frction of the whole circumference. Length of rc p 3 r Are of circles An re is the mount of surfce tht covers closed shpe. To find the re of circle, complete the following investigtion. I N V E S T I G A T I O N Are of sector Wht you will need: crdord, pir of compsses, pencil nd ruler. Wht to do 1 Drw circle of rdius 6 cm on piece of crdord. Cut this circle into eight sectors, s you would do for pizz. 3 Cut the lst sector into hlf gin. 4 Arrnge the seven sectors lterntely, s shown in the digrm Attch the hlves of the lst sector to either end to form rectngle. Note tht this rectngle is mde up of the whole surfce of the circle. In generl, if the rdius of the circles is r cm, then: length of the rectngle 5 hlf of the of the circle 5 p width of the rectngle 5 of the circle 5 re of the rectngle 5 length 3 width re of the circle (units) From the investigtion, the re of the circle 5 pr (units). The re of sector is frction of the re of the whole circle. Hence: Are of the sector p 3 r

18 9 oxford mthemtics for victori Construction of 90 By following these instructions, right ngle ( 90 ngle) cn e constructed using ruler nd pir of compsses. Step 1 Drw line AB tht mesures 4 cm. A B Step With A s centre nd the rdius little more thn hlf of AB, drw semicircle to cut AB t C nd AB extended t D. D A C B Step 3 With C s centre nd the rdius little more thn AC, drw n rc ove the semicircle. D A C B With D s centre nd using the sme rdius, cut this rc t E. Step 4 Drw line joining A to E, using ruler. Angle EAB is 90. E Step 5 D A C B Constructing tngent to circle We know tht tngent touches circle t one point nd the rdius mkes right ngle with the tngent t tht point. Step 1 Drw circle of rdius 3 cm. Drw the rdius OA of the circle. A O Step With A s centre nd rdius little more thn hlf of OA, drw semicircle to cut OA t B nd OA extended t C. C A B O

19 re nd volume 93 With B s centre nd rdius little more thn AB, drw n rc ove the semicircle. Step 3 C A B O With C s centre nd using the sme rdius, cut this rc t D. D Step 4 C A B O Step 5 Drw line joining A nd D. Extend the line. Angle DAO is 90 nd the line AD is the tngent t the point A. C D A B O E X A M P L E S Exmple B1 Find the circumference of circle with rdius.5 cm. Find the circumference of circle with dimeter 7.3 cm Give your nswer correct to deciml plces. (Use p ) Circumference of circle 5 pr or pd (since D 5 r). Use the formule. Circumference 5 pr cm cm Circumference 5 pd cm 5.9 cm Exmples continued next pge

20 94 oxford mthemtics for victori Exmples continued Exmple B Find the rdius of circle with circumference 154 cm. Find the dimeter of circle with circumference 15 mm. Give your nswers to deciml plces. (Use p ) C 5 pr, hence r 5 p C C 5 pd, hence D 5 p C Use the formule. Rdius 5 C (p) ( ) cm cm Dimeter 5 C p mm mm Exmple B3 Find the re of the following circles with: i rdius cm ii dimeter cm Find the rdius of circle if the re is cm. i Are of circle 5 pr ii Rdius 5 D Rdius of circle 5 p A Use the formule. i Are 5 pr cm cm ii Rdius cm Are cm cm Rdius 5 A p cm 3.14

21 re nd volume 95 Exmple B4 For the following sectors, determine: i the frction of the circle ii the length of the rc iii the re of the sector cm i Frction of circle ii The length of n rc is frction of the whole circumference. iii The re of sector is frction of the re of the whole circle. Use the formule. i Frction of circle ii Length of rc p 3 r cm cm iii Are of sector p 3 r cm cm i Frction of circle ii Length of rc cm 4 iii Are of sector cm Exmples continued next pge

22 96 oxford mthemtics for victori Exmples continued Exmple B5 For the following shpes, find: ii the perimeter 7 ii the re. 7 cm 9 cm 15 cm Frction of circle Length of rc of semicircle is C 5 pr 4 5 pr Are of semicircle 5 pr i Find the length of the rc, nd dd the one rdius to the other three sides of the squre. length of rc p 3 r cm cm perimeter of shpe cm cm cm ii Find the re of the sector nd the re of the squre nd dd them. Are of shpe p 3 r 1 l 5 ( ) cm cm i Find the length of the rc of the semicircle nd dd it to the length of the three sides of the rectngle. Rdius of semicircle cm length of rc of semicircle 5 pr cm cm perimeter of shpe cm ii Find the re of semicircle nd re of rectngle nd dd them Are of shpe cm

23 re nd volume 97 Exmple B6 Annek jogs round the inner side of running trck, wheres Clvin jogs round the outer side of the trck. Wht distnce does Annek jog? Wht distnce does Clvin jog? c Wht is the difference etween the distnces long the two edges of the trck? d Find the re of the trck (ie the white re). The circumference of inner circle C 5 pd. Find the circumference of inner circle nd dd this result to the inner stright edges. Do the sme for the outer circle nd stright edges. Circumference of inner circle C 1 5 pd m m Length of inner edge of trck m The distnce Annek jogs m. Circumference of outer circle, C m m Length of outer edge of trck m The distnce Clvin jogs m. c The difference in the distnce long the two edges 5 ( ) m m d To find the re of the trck, we need to find the re of the inner shpe nd outer shpe nd sutrct the first from the second. Are of inner shpe 5 ( ) m m Are of outer shpe 5 ( ) m m Are of trck (white section) m Exercise 3B 1 Nme the mrked fetures of these circles. c

24 98 oxford mthemtics for victori d e f g h i j k l For the circles illustrted, clculte i the circumference, ii the re. Give your nswers correct to deciml plces. (Use p ) c 3. cm 8 cm 7. m d e f 18.3 cm 6.7 m 3 Using the circumference given, determine: i the dimeter ii the rdius. Give your nswers correct to deciml plces. 15 cm 75 cm c 39 cm d 11.4 cm e 13.7 cm f 75.7 cm 4 For ech of the following sectors, determine: i wht frction of the circle it represents ii the length of rc (correct to deciml points where necessry) iii the perimeter of the shpe (correct to deciml plces where necessry) iv the re of the shpe (correct to deciml points where necessry). c cm 7. cm 3.7 cm

25 re nd volume 99 d e f mm 11.6 cm cm 5 For the following res of circles, determine: i the rdius ii the dimeter Give your nswers correct to deciml plces. 100 cm 54 cm c 18 cm d 56.7 cm e 34 cm f 576 cm 6 For the following composite shpes, clculte: i the perimeter ii the re Give your nswers correct to deciml plces: c 8.5 cm 1.4 mm 39 mm 9. cm 3.6 mm 1 cm d e f 13 m cm 7.7 cm g 5.3 cm h 7 10 cm 8 cm 6 cm 5.7 cm 7 Determine the shded re in ech of the following, correct to deciml plces. c 4 cm 6 m 8 m

26 100 oxford mthemtics for victori d e f 7.4 cm 4.4 cm 7 m 1.6 cm 8 A tyre completes 64 revolutions to trvel 1 km. Clculte the distnce trvelled y the tyre in one revolution. Clculte the rdius of the cr s wheel, correct to the nerest centimetre. 9 A rectngulr yrd hs swimming pool surrounded y pving. Determine the re occupied y the swimming pool. Give your nswer correct to the nerest m. (Use p ) Correct to the nerest m, determine the re to e pved. c If pving tile mesures mm, find the numer of tiles required to pve the remining yrd. d If ech tile costs $5.50 nd the lour cost of pving is $30 per squre metre, find the totl cost of pving. Include 10% GST on oth costs.) 10 Two htoxes hve the sme se re. One hs rectngulr se with width of 11.5 cm, nd the other hs circulr se of rdius 15.4 cm. Clculte the length of the rectngulr se. 11 Flowereds re designed inside rectngulr grden, s shown in the figure. The rest of the re is covered with lwn. Find the re of ll the flowereds. Find the re of the lwn. 1 Semicircles re drwn on ech side of rightngled tringles, s given in the digrm. Clculte the re of ech semicircle. In ech cse, find reltionship etween the lrge semicircle nd the two smller semicircles. c If qudrnts re drwn on the sides, find the reltionship etween the lrge qudrnt nd the two smller qudrnts. 5 cm 7 cm 4 cm

27 re nd volume Construct tngent t ny point on circle with: rdius 5.5 cm rdius 7 cm c rdius 6.3 cm 3C Surfce re of solids using nets Homework Sheet 3.1 A solid is nything tht tkes up spce. Dotted lines re used to give the impression tht the oject is three-dimensionl tht it hs three dimensions: length, width nd height. Exmples of solids re shown. cue cylinder cone The flt or curved section of solid is clled the surfce or fce. A cue hs six fces nd ech is in the shpe of squre. A cylinder hs two flt surfces in the shpe of circles nd one curved surfce when opened out hs the shpe of rectngle. A cone hs one flt surfce in the shpe of circle nd one curved surfce tht when opened out is in the shpe of sector of circle. We will lern more out these solids lter. Types of solids Polyhedrons Any solid tht is mde up of ll flt surfces is clled polyhedron (the plurl form is polyhedr). Ech flt surfce is clled fce nd is in the shpe of polygon A polygon is closed shpe mde up of stright lines. Some exmples re the tringle, the squre nd the hexgon. The corner point in polyhedron is clled the vertex. The intersection of two flt surfces is clled n edge. In the digrm, ABEF is fce nd A, B, C, etc re vertices (the plurl form of vertex). Length CG is n edge. Pltonic solids There re five regulr polyhedr tht re clled Pltonic solids. They re nmed fter the Greek philosopher, Plto. These solids hve een known for t lest 3000 yers. In regulr polyhedron: ll fces re congruent (the sme size nd shpe) ll edges hve the sme length ll vertices hve the sme numer of edges meeting t them ll fces re regulr polygons.

28 10 oxford mthemtics for victori The five regulr polyhedr re descried in this chrt. Tetrhedron 4 tringulr fces 3 tringulr fces meeting t vertex Cue 6 squre fces 3 squre fces meeting t vertex Octhedron 8 tringulr fces 4 tringulr fces meeting t vertex Dodechedron 1 pentgonl fces 3 pentgonl fces meeting t vertex Icoshedron 0 tringulr fces 5 tringulr fces meeting t vertex Prisms A prism is solid hving two congruent fces, eing the rest of the fces re rectngles. The congruent fces re the sme shpe s the cross-section. A cross-section is slice of the solid nd is cut prllel to its congruent fce. Both congruent fces re clled the se of the prism. The se cn e ny polygon. The nme of the prism depends on the shpe of its se. If the rest of the fces re rectngles, then the prism is clled right prism. Shown here re some prisms. cylinder hexgonl prism cue tringulr prism rectngulr prism

29 re nd volume 103 Some exmples of prisms re shown here. cylinder cuoid or rectngulrsed prism cue cross-section (slice) cross-section (slice) cross-section (slice) or tringulr-sed prism hexgonl-sed prism cross-section (slice) cross-section (slice) Pyrmids A pyrmid is solid with se in the shpe of polygon; the rest of the fces re tringles meeting t point clled the pex or vertex. The cross-sections of pyrmid re the sme shpe s the se ut re of different sizes. A tpered solid tht hs its pex ove the centre of the se is clled right pyrmid or right cone. A cone is solid tht hs circulr se nd curved surfce tht tpers to point clled the pex.

30 104 oxford mthemtics for victori The sphere nd the hemisphere Other solids re the sphere nd the hemisphere. A sphere is solid tht hs one curved surfce. It is like ll. A cross-section pssing through the centre is circle. It hs mny identicl cross-sections tht pss through its centre. A hemisphere is hlf sphere, much like hlf n ornge. It hs one flt surfce in the shpe of circle nd one curved surfce. Sphere Solid Digrm Cross-sections through the solid cross-section 1 cross-section cross-section 1 cross-section Hemisphere cross-section 1 cross-section cross-section 1 cross-section Surfce re using the net of solid A net of solid is digrm showing ll its fces. If ox in the shpe of cue is opened out nd lid flt, it will look like this digrm. The digrm is the net of the cue. It shows six squre fces. The totl surfce re or, simply, the surfce re, is the sum of the res of the six squres. If the side of the cue is 3 cm, then: Surfce re of cue re of squre cm To find the surfce re of solid using its net: drw the net of the solid find the re of ech fce dd the res of the fces.

31 re nd volume 105 E X A M P L E S Exmple C1 For ech solid: ii drw the cross-section ii identify the numer of fces. The cross-section is slice cut prllel to the congruent se. The numer of fces is the numer of surfces tht cn e seen. i ii There re pentgonl fces nd 5 rectngulr fces: 7 fces. i ii There re tringulr fces nd 3 rectngulr fces: 5 fces. Exmple C Find the surfce re of the rectngulr prism, using net. 48 cm 96 cm 3 cm Surfce re is the sum of the res of the fces tht mke up the net. Drw the net. Add the res of ll the fces. 3 cm 96 cm 96 cm 48 cm 48 cm The surfce re of the rectngulr prism 5 (96) 1 (48) 1(3) cm 3 cm Exmples continued next pge

32 106 oxford mthemtics for victori Exmples continued Exmple C3 Find the surfce re of the rectngulr prism, using its net. 3 cm 8 cm The net hs two squre fces tht re the sme nd four rectngulr fces tht re the sme. Are of squre 5 l Are of rectngle 5 l 3 w Drw the net. Find the re of ech fce nd dd them. Are of squre ses 5 (3) 5 18 cm Are of rectngulr fces 5 4(8 3 3) 5 96 cm 3 cm 8 cm The surfce re of the rectngulr prism 5 (3) 1 4(8 3 3) 5 ( ) cm cm Exmple C4 Find the surfce re of the tringulr prism, using its net. The net of the tringulr prism is mde up of two tringles tht re the sme nd two rectngles tht re the sme. The third rectngle is different. Drw the net. Find the res of the two tringles nd three rectngles nd dd them.

33 re nd volume 107 Exmple C4 Continued Are of two tringles 5 ( 1_ ) 5 4 cm Are of two smll rectngles 5 (5 3 1) 5 10 cm Are of lrger rectngle cm Surfce re of the tringulr prism 5 ( 1_ ) 1 (5 3 1) 1(6 3 1) 5 ( ) cm 5 16 cm Exmple C5 Find the totl surfce re of rectngulr pyrmid cm 8.74 cm 9 cm 6.5 cm The net of rectngulr pyrmid is mde up of four tringles nd one rectngle. Drw the net. Find the re of ech fce nd dd them. Are of two smll tringles 5 ( 1_ ) cm Are of two lrger tringles 5 ( 1_ ) cm Are of rectngulr se cm 6.5 cm 9 cm 9.54 cm Surfce re of the rectngulr pyrmid 5 ( 1_ ) 1 ( 1_ ) 1 ( ) cm 5 ( ) cm cm 8.74 cm

34 108 oxford mthemtics for victori Exercise 3C 1 For ech solid, find the numer of: i fces ii edges iii vertices c From these digrms, identify those tht re: i prisms ii pyrmids c d e f g h i j k l m n o p q r s t 3 Sketch cross-section of ech solid prllel to its se. c

35 re nd volume 109 d e f g h 4 For ech net: i clculte the surfce re ii drw the shpe of the solid formed. 5 mm 70 mm 70 mm 70 mm 5 mm c 90.8 cm d 5.5 cm ech 35.6 cm 90.8 cm cm 90.8 cm 5.5 cm 90.8 cm e 8 mm f ech is 19.3 m mm mm mm mm 8 mm 5 For ech prism: i drw the net 36 cm ii find the surfce re. top 108 cm side 70 cm 36 cm front 80 cm

36 110 oxford mthemtics for victori c sloping fce 350 cm d ottom 170 cm tringulr se 54 cm e f 49 cm sides 11.7 cm g ech fce 76 cm h 45.8 mm 150 mm 9 cm 6 For ech prism: i drw the net ii find the surfce re. 4 cm 9 cm 1 cm 10 cm 6 cm 11 cm c 3 cm d 4 mm 5. cm 1.5 cm 6. mm 8.3 mm e 30 mm f 1.7 m 35 mm 5 mm 7 mm 40 mm 153 m 170 m g 5 m h 6 cm 4 cm 4 m 7 m 3 m 8 cm 10 cm

37 re nd volume 111 i j 7 For ech pyrmid: i drw the net ii find the surfce re. c d 8 Find the surfce res of the solids formed y these nets. c d

38 11 oxford mthemtics for victori 9 For ech prism: i find the re of the front fce ii find the totl surfce re. c d e f Ignore the inside surfce re of this prism. 10 A squre prism hs n re of 348 cm. The squre fce hs side of 6 cm. Find the other dimension. How mny rectngulr oxes mesuring 1 cm y 16 cm y 5 cm could e covered, with no overlpping, with 6 m of plstic wrp? 3D Surfce re of solids using formule As seen in the previous section, the surfce re of solid cn e otined y using the net tht mkes up the outer surfces of the solid. Using this ide, we cn otin formule to find the surfce res of certin solids. The surfce re of cue is mde up of six identicl squres. If the side of cue is x cm, then the totl surfce re of cue re of one squre. Totl surfce re of cue 5 6x

39 re nd volume 113 The surfce re of cuoid or rectngulr prism is mde up of three pirs of congruent rectngles; tht is, totl of six rectngles. Consider the rectngulr prism nd its net. The res of the three pirs of rectngles re: l 3 w 5 lw w 3 h 5 wh h 3 l 5 hl The surfce re of the rectngulr prism 5 lw 1 wh 1 hl. Surfce re of rectngulr prism 5 (lw 1 wh 1 hl) The cylinder hs two congruent circulr surfces nd one curved surfce. When opened out, the curved surfce forms rectngle, nd its length is the circumference of the circulr se. The width of the rectngle is the height of the cylinder, s seen in the digrms elow. As seen from the net, the curved surfce re of the cylinder 5 re of the rectngle 5 lxw 5 pr 3 h. Curved surfce re of cylinder 5 prh The totl surfce re of the cylinder is mde up of the res of the two circles nd the curved surfce re. So the totl surfce re of the cylinder 5 prh 1 pr. Rememer Are of circle 5 pr Totl surfce re of cylinder 5 pr (h 1 r) It is not possile t this stge to find the formule for the surfce re of every type of solid. If we need to find the surfce re of ny other type of solid, we must find the re of ll the surfces exposed (ie tht cn e seen) nd dd them.

40 114 oxford mthemtics for victori E X A M P L E S Exmple D1 Find the surfce re of cue with sides 4.3 cm. Surfce re of cue 5 6x. Use the formul. Surfce re 5 6x 5 6(4.3) cm cm Exmple D Find the totl surfce re of rectngulr prism with length 7.3 cm, width 6. cm nd height.5 cm. Surfce re of rectngulr prism 5 (lw 1 wh 1 hl), Use the formul. Surfce re 5 (lw 1 wh 1 hl) 5 ( ) cm 5 ( ) cm cm cm Exmple D3 7 cm Find the curved surfce nd the totl surfce re of cylinder with rdius 7 cm nd height 10.5 cm. (Use p ) Give your nswer correct to deciml plces cm The curved surfce re of cylinder 5 prh Totl surfce re of cylinder 5 pr(h 1 r). Use the formule. Curved surfce re 5 prh cm cm Totl surfce re 5 pr(h 1 r) ( ) cm cm cm

41 re nd volume 115 Exmple D4 Find the surfce re of the tringulr prism illustrted. 3 cm 8 cm 5 cm 11 cm The tringulr prism hs two congruent tringulr surfces nd three rectngulr surfces. To find the surfce re of composite shpes, we find nd dd the res of ll the surfces tht cn e seen. Find the re of the two tringles nd the re of the rectngles nd dd them. Are of tringle 5 3 h m Are of side rectngle 5 l 3 w m Are of ottom rectngle m Totl surfce re of tringulr prism 5 ( ) m 5 ( ) m 5 m Exmple D5 Find the surfce re of this composite solid. The front fce is mde up of trpezium nd rectngle. There re two such fces. The rest of the fces re rectngles. Find the re of ll the fces nd dd them. Are of trpezium fces 5 [ 1_ ( 1 )h ] 5 [ 1_ (14 1 8) 3 6 ] 5 1() m Are of front nd ck rectngles 5 lw m Are of two slnting rectngles m Are of two side rectngles m Are of top rectngle m Are of ottom rectngle m Hence, the surfce re of the composite solid 5 ( ) m. Exmples continued next pge

42 116 oxford mthemtics for victori Exmples continued Exmple D6 A shed hs to e pinted with protective sustnce to withstnd the hrsh wether conditions. Find the surfce re tht needs to e pinted. The door need not e pinted with this sustnce. If the cost of 5 L cn of pint is $57.95 nd one cn covers 6 m, find the numer of cns required to complete the jo, nd the cost of pinting the whole shed. The surfce to e pinted is mde up of two tringles nd numer of rectngles. Exercise 3D 1 Use the formul to clculte the surfce re of ech cue. c d 11 cm 0.6 m 15.7 cm 0.45 m Use the formul to clculte the surfce re of ech rectngulr prism. Give your nswer to the nerest squre metre. 30 cm 7 m 74 cm 100 cm 4.1 m 6.3 m

43 re nd volume Clculte the surfce res of the following tringulr pyrmids using the pproprite formul for ech fce. 6 m 1 cm 10 cm 17 m 3 m 15 cm 18 cm 4 m c d 18.5 cm 4.6 cm 3 cm.3 cm 17.3 cm 8 cm 5.8 cm 14.5 cm e f 1.7 cm 18 cm 8 cm 4 cm 7. cm 14.6 cm 11.3 cm 4 For ech cylinder, use the pproprite formule to find: i the curved surfce re ii the totl surfce re Give your nswers to the nerest cm. Use p c d 8.4 m e f 0.75 m 7.5 m 0.3 m 1.5 m 0.3 m 5 Clculte the surfce res of these composite solids. 1.5 m 3 m m 3. m.5 m Ignore the inside surfce re of this solid.

44 118 oxford mthemtics for victori c 15 cm d 9.5 cm 4 cm 5 cm 18.5 cm 6 The wlls nd ceiling of edroom re to e pinted. Find the surfce re of the four wlls (excluding the door nd the windows) nd the ceiling. The dimensions of the door re.1 m 3 90 cm nd ech window is 1.8 m 3 90 cm. Give your nswer to the nerest squre metre. Clculte the cost of pinting if 3 L cn costing $5.95 covers n re of 7 m. 3. m 4.8 m.6 m 7 The digrm shows tent. Clculte the smllest mount of fric required to mke this tent if n extr 6% llownce is required for sems.. m.5 m 3 m 3. m 8 Wht is the externl surfce re of the illustrted crdord luminium foil roll? Homework Sheet 3. 3E Volume of solids The volume of solid is the mount of spce it occupies, or the mount of mteril tht mkes up the solid. It is mesured in cuic units. In order to understnd this concept, we could crry out n investigtion. I N V E S T I G A T I O N Volume of rectngulr prisms 1 Clculte the volume of this lyer of cues. 5 cm 4 cm 1 cm Wht will e the volume of the rectngulr prism formed y stck of: i 3 lyers? ii 5 lyers? iii 10 lyers?

45 re nd volume 119 This rectngulr prism is formed y stcking lyers of cues on top of one nother. How mny cuic centimetres re there in the ottom lyer? How mny lyers re in the stck? c Wht is the volume of this prism? 3 Repet Question for the rectngulr prisms given elow. cm cm 6 cm c d 3 cm 3 cm 4 cm 6 cm cm 5 cm 4 Using the results of questions nd 3, write in words nd symols rule for clculting the volume of rectngulr prism. Discuss with other groups nd modify your rule if you wish to do so. 4 cm 5 cm 3 cm From the investigtion ove, it is oserved tht: Volume of rectngulr prism 5 l 3 w 3 h where l 5 length of rectngulr prism w 5 width of rectngulr prism nd h 5 height of rectngulr prism. Clculting volumes for other solids In the investigtion ove, we oserved tht the first lyer is the se of the rectngulr prism. This is lso referred to s the cross-section of the prism. We lernt in the lst section tht prism hs uniform cross-sections if the slices cut prllel to the se re the sme.

46 10 oxford mthemtics for victori Hence: Are of se 5 Are of cross-section 5 l 3 w. So, the volume of rectngulr prism 5 l 3 w 3 h 5 re of se 3 height. Hence, for ny prism tht hs uniform cross-section s se, the generl formul to find its volume is: Volume of prism 5 re of se 3 height If solid hs se tht is n irregulr shpe, the volume is found y using the generl formul for prism. The re of the se will e supplied. Formule to find the volume of prisms If the se re is not given, it should e found using the pproprite formul. In the previous section we studied the re of vrious regulr polygons, which will help us in this section. Cue Are of se of cue of side x 5 x (unit) Height of cue 5 x (unit) Hence: Volume of cue 5 re of se 3 height 5 x 3 x 5 x 3 Volume of cue 5 x 3 x Note Sometimes the prism might e in horizontl position. In such cses the height will e its length, ut we still use the word height in the formul. x x Tringulr prism Are of (se) tringle h Height of tringulr prism 5 H Hence: Volume of tringulr prism 5 1_ 3 3 h 3 H Cylinder Are of (se) circle 5 pr h r H Note h nd H stnd for different prts of the prism. Height of cylinder 5 h Hence: h Volume of cylinder 5 pr h Other prisms It is not possile to get formule for the volumes of ll the different types of prisms. However, it is possile to find volumes y using the generl formul for the volume of prism. If the se of the prism is composite shpe, then the re of this se is found y clculting the re of ech regulr shpe nd dding or sutrcting them.

47 re nd volume 11 Volume of tpered solids Tpered solids re solids with fces tht meet t point. Exmples include the cone nd the pyrmid. I N V E S T I G A T I O N Volumes of solids nd pyrmids You will need: crdord, sticky tpe nd scissors. Wht to do 1 Construct nd cut out the nets of the open cue nd the open squre pyrmid elow using the informtion on the digrms. The cue nd the pyrmid hve the sme se nd height mesurements. Use sticky tpe to mke the ojects. Fill the pyrmid with snd nd see how mny pyrmids of snd it tkes to fill the cue. Construct others of your own nd test them. Fill the cone with snd nd see how mny cones of snd it tkes to fill the cylinder. Construct others of your own nd test them. 94 mm open cylinder 15 mm open cone 0 mm 16 0 mm open pyrmid with squre se It is oserved tht the cue is filled with three times the mount of snd in the squre pyrmid nd the cylinder is filled with three times the mount of snd in the cone. This is provided tht the solids hve the sme se nd height. Hence: The volume of the tpered solid 5 1 (volume of the prism) mm 5 mm Volume of tpered solid 5 1_ 3 re of se 3 height of pyrmid 3 Construct nd cut out the nets of the open cylinder nd the open cone using the informtion on the digrms. The cylinder nd the cone hve the sme circulr se nd height mesurements. Use sticky tpe to mke the ojects. Hence: Volume of squre pyrmid 5 1_ 3 x h Volume of cone 5 1_ 3 pr h

48 1 oxford mthemtics for victori Composite solids The volume of composite solid is found y clculting the volume of ech solid contined in it nd then dding or sutrcting them. E X A M P L E S Exmple E1 Find the volume of this prism. 7 cm 11 cm 6 cm Volume of rectngulr prism 5 l 3 w 3 h, where length of prism 5 11 cm width of prism 5 6 cm height of prism 5 7 cm. Use the formul. Volume of prism 5 l 3 w 3 h cm 3 Exmple E Find the volume of these prisms. The re of ech se is given. 1 cm 40 cm 7 cm 15 cm Are of se 5 40 cm, height 5 1 cm Are of se 5 7 cm, height 5 15 cm Use the formul. Volume of rectngulr prism 5 re of se 3 height cm 3 Volume of pentgonl prism 5 re of se 3 height cm 3

49 re nd volume 13 Exmple E3 Using p , clculte the volume of the following prisms: c 3. cm 7 cm Give your nswers correct to deciml plces cm Volume of cue 5 x 3 Volume of tringulr prism 5 1_ 3 3 h 3 H Volume of cylinder 5 pr h Use the formule. Volume of cue 5 x 3 5 (7) cm 3 Volume of tringulr prism 5 1_ 3 3 h 3 H 5 ( ) cm cm 3 c Volume of cylinder 5 pr h 5 ( (3.) ) cm cm 3 Exmple E4 Find the volume of the trpezoidl prism. 17 m 11 m 4 m 1 m The se is trpezium with prllel sides mesuring 17 m nd 11 m. The perpendiculr distnce etween them is 4 m. Height of prism is 1 m. Use the pproprite formule. Are of (se) trpezium 5 1_ 3 h( 1 ) ( ) 5 56m [ Volume of trpezoidl prism 5 re of se 3 height m 3 Exmples continued next pge

50 14 oxford mthemtics for victori Exmples continued Exmple E5 Clculte the volume of this prism. 5 m 14 m 11 m 15 m The se of the prism is mde of rectngle nd tringle. Height of the prism 5 15 m. Find the re of the tringle nd the rectngle nd dd them to give the re of the se of the prism. Are of se 5 1_ h 1 lw 5 ( 1_ ) m 5 ( ) m [ Volume of prism 5 re of se 3 height m 3 The volume of this solid could lso e found y finding the volumes of the rectngulr prism nd tringulr prism nd dding them. Exmple E6 A rectngulr piece of metl mesuring 115 cm y 144 cm is rolled long its longer side to form hollow cylinder. Find the volume, correct to deciml plces, of the resulting cylinder. (Use p ) 115 cm 144 cm 115 cm 144 cm The width of the rectngle is the circumference of the se of the cylinder. The height of the cylinder is the length of the rectngle. Rdius of cylinder 5 C (p) cm ( ) [ Volume of cylinder 5 pr h 5 ( ) cm 3

51 re nd volume 15 Exmple E7 Find the volume of the following solids. Side of squre se 5 4 cm Height of pyrmid 5 1 cm Rdius of circulr se of cone 5 7 cm Height of cone cm Use the formule. Volume of squre pyrmid 5 1_ 3 x h Volume of cone 5 1_ 3 pr h 5 1_ cm 3 5 1_ cm 3 Exercise 3E 1 Find the volume of ech solid, given the re of the se. Give your nswers to deciml plces where necessry. c 4 cm 10 cm 11 cm 54.6 cm d e f 5 m 15 h 44 m 10 m g

52 16 oxford mthemtics for victori Using pproprite formule, find the volume of ech solid. Give your nswers correct to deciml plces. (Use p ) c 55 mm 1.3 cm 1.5 cm 4 cm 11.6 cm 7.6 cm 9 cm d e f 9.4 cm 3.5 cm 8 cm 140 mm 4 m 30 m 175 mm 7 m g 14.6 cm h 146 mm 3 Find the volume of ech solid. Give your nswers correct to deciml plces. (Use p ) 17 cm 60 7 cm 18 cm 5 cm 1 cm 6 cm c d 7.6 cm 17.3 cm 5 mm e f 7.7 cm 144 mm 18.6 cm 7. cm 140 mm 9.6 cm

53 re nd volume 17 4 Find the volume of ech tpered solid. Give your nswers correct to deciml plces. Use p cm c 11 cm 7.5 cm 3 cm d e f 7.6 cm 6.7 cm 115 mm 7.4 cm 8. cm 5 Find the volume of ech composite solid. Give your nswers correct to deciml plces. Use p cm 8.6 cm 11.6 cm c d.5 cm 3.7 cm 1 cm 8.3 cm 4.8 cm 7.5 cm 8.7 cm 6 A cylinder of dimeter 180 mm nd height 10 mm is melted nd cst into rectngulr rs mesuring 0 mm 3 30 mm 3 40 mm. How mny rs cn e produced? 7 A rectngulr sheet of crdord mesuring 4 m 3 36 m is rolled on its shorter end to form cylinder. Find: the rdius of the cylinder the volume of the cylinder.

54 18 oxford mthemtics for victori 3F Cpcity Cpcity is the mesure of spce inside solid continer. The spce cn e filled with fluid (liquid or gs). Hence, cpcity is the volume of fluid tht solid continer cn hold. The unit of cpcity depends on the units for volume. The units for cpcity re the millilitre (ml), litre (L), kilolitre (kl) nd meglitre (ML). The sic unit for cpcity is the litre. For esy conversion, digrm is given here. Rememer 1000 ml 5 1 L 1000 L 5 1 kl 1000 kl 5 1 ML ML kl L ml Some fmilir cpcities re given in this tle. Item Medicine glss Cup Milk crton Petrol tnk Hotwter system Estimte of cpcity 5 ml 50 ml 1 L 65 L 170 L 50 m swimming pool 1500 kl Dm Reservoir 10 ML 1000 ML Volume conversion Using the unit of conversion for length, we cn find the unit of conversion for volume. We know tht 1 cm 5 10 mm nd 1 cm 3 1 cm 3 1 cm 5 10 mm 3 10 mm 3 10 mm Hence: 1 cm mm 3 We know tht nd Hence: 1 m cm 1 m 3 1 m 3 1 m cm cm cm 1 m cm cm 3

55 re nd volume 19 A conversion digrm for esy clcultion is given here m 3 cm 3 mm Cpcity/volume conversion If we fill cue of sides 1 cm with wter nd pour the wter into mesuring cup, it will red s 1 millilitre (1 ml). The volume of wter is the sme s the volume of the cue tht is, 1 cm 3. Hence: 1 cm ml 3 oth sides y 1000, 1000 cm ml 1 cm 1 ml 1000 cm L (since 1000 ml 5 1 L) 3 oth sides y 1000, cm L ( cm m 3 ) 1 m L or 1 m kl (since 1000 L 5 1 kl) E X A M P L E S Exmple F1 Convert the following m 3 to cm 3.8 cm 3 to mm 3 c 7500 mm 3 to cm 3 d cm 3 to m 3 1 m cm 3 1 cm mm 3 Use the conversion digrm m 3 5 ( ) cm cm 3.8 cm 3 5 ( ) mm mm 3 c mm 3 5 ( ) cm cm 3 d cm 3 5 ( ) m m 3 Exmples continued next pge

56 130 oxford mthemtics for victori Exmples continued Exmple F Convert the following ml to L 30 kl to L c 1500 kl to ML d 3.5 L to ml 1000 ml 5 1 L 1000 L 5 1 kl 1000 kl 5 1 ML Use the conversion digrm ml 5 ( ) L 30 kl 5 ( ) L c 1500 kl 5 ( ) ML d 3.5 L 5 ( ) ml Exmple F3 Find the cpcity of continers with the following volumes cm m 3 c 4570 mm 3 1 cm ml 1 m L 1 m kl 1 cm mm cm ml 5 ( ) 5 4 L 3.7 m 3 5 ( ) L kl c 4570 mm 3 5 ( ) cm cm ml Exmple F4 Find the cpcity of the given continers. 6.4 cm 17.5 cm 3 cm 5 cm 17 cm 18 cm Volume of tringulr prism 5 1_ hh Volume of rectngulr prism 5 lwh 1 cm ml 1 L ML Use the formule. Volume of tringulr prism 5 1_ 3 3 h 3 H 5 ( ) cm cm ml 5 ( ) L Volume of rectngulr prism 5 l 3 w 3 h 5 ( )5 979 cm ml 5 ( ) L

57 re nd volume 131 Exmple F5 Find the cpcity of the cylindricl cn. If one mug holds 50 ml of liquid, how mny mugs must e used to fill the cn? (Use p 5 7.) Volume of cylinder 5 pr h r cm, h 5 18 cm 1 cm 3 5 1mL Use the formule. Volume of cylindricl cn 5 pr h 5 ( ) 5 77 cm 3 Cpcity of cn 5 77 ml 5.77 L Numer of mugs used 5 ( ) 11 mugs Exmple F6 A fish tnk mesuring 30 cm y 5 cm y 15 cm is filled with wter to depth of x cm. If the volume of the wter is 9 L, find the depth of wter in the tnk. The volume of the rectngulr tnk to depth of x cm 5 volume of wter in the tnk. Convert to sme units: Volume of wter in tnk 5 9 L ml cm 3 Form n eqution to solve for the unknown, x. Volume of tnk 5 volume of wter x x x cm Depth of wter 5 1 cm Exercise 3F 1 Convert ech of the following to the unit given. 5 cm 3 5 mm cm 3 5 m 3 c 0.83 cm 3 5 mm 3 d.86 m 3 5 kl e 575 cm 3 5 L f 683 L 5 cm 3 g 4 m 3 5 cm 3 5 L h 6000 L 5 m 3 5 cm 3 i cm 3 5 m 3 5 kl j 0.5 kl 5 L 5 cm 3

58 13 oxford mthemtics for victori For ech continer, find: i the volume in cuic centimetres iii the cpcity in litres. 15 cm ii the cpcity in millilitres c d 3 A 4 cm circulr cke tin hs depth of 7 cm. Find, in millilitres, the volume of cke mixture it cn hold if it is three-qurters full. 4 A concrete sl mesuring 3.5 m y.7 m y 1.6 m is to e lid. Find the volume of concrete used to ly this sl. If the cost of concrete is $180 per cuic metre, find the totl cost of lying the sl. 5 A fridge hs the following internl dimensions: height 5 15 cm, width 5 90 cm nd depth 5 50 cm. Find the cpcity of the fridge in litres. 6 A swimming pool, s shown in the digrm, hs length of 0 m nd width of 8 m. The depth is 1. m t the shllow end nd.4 m t the deep end. Nme the shpe of the cross-section of the pool. Find the volume of the pool. c Find the cpcity of the pool in kilolitres. d If the pool is filled to depth of 90 cm, find the volume of wter in kilolitres. 7 The volume of the lock of wood is the sme s the cpcity of the te light. Clculte the volume of the lock of wood. Hence, find the rdius of the te light. 4 cm 3.6 cm Homework Sheet A tringulr hollow prism holds chemicl solution to level of x cm. Find the volume of the chemicl solution in terms of x. If the volume of the solution is 147 cm 3, find the vlue of x. 7 cm 10 cm x cm