1.1 Metric Systems - Miss Johnston's Math...

Section 1.1 - Metric Systems

Copyright © 2009 by Crescent Beach Publishing. No part of this publication may be reproduced without written permission from the publisher.

The International System of Units, abbreviated SI, (from the French le Système International d’unités) is theofficial system of measurement in Canada.

Length

1 kilometre (km) = 1000 metres (m)100 centimetres (cm) = 1 metres (m)10 millimetres (mm) = 1 centimetre (cm)

Conversions

1 km = 1000 m or 1 m = 0.001 km100 cm = 1 m or 1 cm = 0.01 m10 mm = 1 cm or 1 mm = 0.1 cm

The advantage of the metric system is that it is easy to convert from one unit to another. That is because the metric system is based on the number 10.

The basic unit of measure is the metre. The metre is used for expressing the dimensions of larger objects, such as the height of a building. The millimetre is used to measure small distances, especially in science andindustry, such as the thickness of a pane of glass. The kilometre is used for large distances, such as the distance between two cities.

Find the missing lengths.

a) 1 km = cm

b) 12.3 m = mm

►Solution: a) 1 km = 1000 m, 1 m = 100 cm

then 1000 m = 1000 # 1 m = 1000 # 100 cm = 100 000 cm

b) 1 m = 100 cm, 1 cm = 10 mm

then 12.3 m = 12.3 # 1 m = 12.3 # 100 cm = 1230 cm

and 1230 cm = 1230 # 1 cm = 1230 # 10 mm = 12 300 mm

Example 1

Metric Systems1.1

♦ 5Lambrick Park Secondary



1.1 Exercise Set

1. Use: mm, cm, m or km to complete the sentence.

a) A ruler is 25 long. b) Vancouver is 2237 from Winnipeg.

c) A loonie is 2 thick. d) The football players ran 7 for a first down.

e) This book is 3 thick. f) The girl is 1.6 tall.

g) The marathon run is 42.3 long. h) The small bug is 8 long.

i) Niagara Falls is 0.053 high. j) A 4 litre milk container is 26 high.

k) A $2.00 coin is 28 in diameter. l) A table is 78 high.

m) A window pane is 8 thick. n) A fir tree is 42 high.

o) A plane is flying at a height of 7.3 . p) Mt. McKinley is 6149 above sea level.




2. Find the equivalent measurement. Do as many as possible mentally.

a) 1 m = cm b) 1 cm = m

c) 1 km = m d) 1 m = km

e) 1 m = mm f) 1 mm = m

g) 1 km = cm h) 1 cm = km

i) 1 km = mm j) 1 mm = km




3. Find the equivalent measurement.

a) 36.3 mm = cm b) 3 km = cm

c) 500 m = mm d) 500 mm = km

e) 8.26 cm = m f) 7.2 mm = m

g) 0.3 km = m h) 0.02 km = mm

i) 63 cm = km j) 4200 mm = km

k) 40 km = mm l) 0.043 km = mm




4. Complete the table.

Object mm cm m kma) Width of sheet of paper

21.7

b) Length of sheet of paper0.28

c) Length of MP3 player92.3

d) Height of CN tower0.553

e) Thickness of phone book3.8 cm

f) Width of library card53.2

g) Length of the Lions Gate Bridge 182300

h) Length of Canadian football field 146.3

i) Thickness of graphic calculator18.2

j) Width of bedroom0.0038




5. Each sentence is incorrect. Move or insert a decimal point to make the sentence correct.

a) The average female height is 1.63 cm.

b) The length of a Persian rug is 0.37 m.

c) A stack of 20 loonies is 391 cm high.

d) The average foot is 2.8 mm long.

e) The highest mountain in Canada is Mt. Logan, Yukon at 59.56 m.

f) The longest river in Canada is the Mackenzie at 4241 m.

g) The highest waterfall in Canada is Della Falls, BC at 4.4 m.

Lambrick Park Secondary



6. Add.

a) 18 m + 1.6 km + 235 cm m

b) 315 mm + 87 cm + 4.2 m cm

c) 0.2 m + 47 cm + 86 mm mm

d) 4.3 km + 64.3 m + 128 cm km

e) 1.8 km + 9.2 m + 7.3 cm + 5.6 mm mm




7. The Trans-Siberian rail is the longest in the world 8. The subway system in New York is 371 000 metres at 944 km. How many centimetres is the track? long, and the subway system in London is 415 km long. How many metres longer is the London system than the New York System?

9. A virus in the human body is 0.000 000 0026 metres 10. The Quingzang railroad in Tibet is the highest in diameter. How many millimetres in diameter is railroad in the world at 5072 metres high. the virus? How many kilometres high is the railroad?

11. The Tour de France bicycle race is 3600 km long 12. The longest certified ultra marathon in the world is and lasts 3 weeks. How many metres are travelled is a race that covers 209 200 metres. How many in the 3 week race? kilometres is this race?




13. The record for the longest swim is 5268 km, set by 14. The record for the longest long jump was set by Martin Strel in the Amazon River over the course Mike Powell of the USA, in 1991. His jump of 66 days. How many kilometres did he swim per measured 8950 mm. How many metres long was day? his jump?

15. The world record for the pole is 6.14 m for men, 16. The world’s tallest man is 246.5 cm in height, and 506 cm for women. How many cm higher is and the world’s shortest man is 0.5715 m in height. the men’s record than the women’s record? What is the difference in their height in cm?


Section 1.2 - Imperial Systems


TheimperialunitofmeasureismainlyusedintheUnitedStates,butinCanadaitisstillusedforcertain measurements.Lumber,plywoodandnailsareexamplesofmanufactureditemsthatuseimperialunits.

Length

12inches(in) = 1foot(ft) 3feet(ft) = 1yard(yd) 36inches(in) = 1yard(yd) 1760yards(yd) = 1mile(mi) 5280feet(ft) = 1mile(mi)

Thissystemofmeasurementismoredifficulttoworkwithcomparedtothemetricsystembecauseitisnot basedonthenumber10.

Convert741 yardstoinches.

►Solution: 1yd=36in

then741 yd=7

41 #1yd=7

41 36# in=261in

Convert57inchestofeet.

►Solution: 12in=1ft

then57inches 57 4ft ft ft121

1257

43#= = =

Convert38fttoyards.

►Solution: 3ft=1yd

then38feet 38 yd yd31

338 12

32#= = = yds

Example 1

Example 2

Example 3

Imperial Systems1.2




Convert17160feettomiles.

►Solution: 5280ft=1mi

then17160 17160 3.25ft mi mi mi52801

528017160#= = =

Convert143 milesto:

a)yardsb)feetc)inches

►Solution: a) 1mi=1760yds

then143 mi= 1

43 1# mi=1

43 1760# yds=3080yds

b) 1yd=3ft

then3080yds=3080#1yd=3080#3ft=9240ft

c) 1ft=12in

then9240ft=9240#1ft=9240#12in=110880in

Convert4inchesto:

a)feetb)yardsc)miles

►Solution: a) 12in=1ft

then4inches 4 1 4in ft121

31# #= = = ft

b) 36in=1yd

then in yd4 1 4 1 136 9

# #= = = yd

c) 5280ft=1mile,12in=1ft

therefore1mile 5280 12 63 360in#= = in

then4inches in mi4 1 4 1 163 360 15 840

# #= = = mi

Example 4

Example 5

Example 6




1.2 Exercise Set

1. Findtheequivalentmeasurement.

a) 1yd=ft b) 1ft=yd

c) 1ft= in d) 1in=ft

e) 1yd=in f) 1in=yd

g) 1mi=yd h) 1yd=mi

i) 1mi=ft j) 1ft=mi




2. Findtheequivalentmeasurement.

a) 28ft=yd b) 96in=ft

c) 4mi=yd d) 10560ft=mi

e) 9ft=in f) 3mi=ft

g) 5yd=in h) 24yd=ft

i) 432in=yd j) 12320yd=mi




3. Findtheequivalentmeasurement.Leaveyouranswerinfractionalform.

a) 10ft=yd b) 66in=ft

c) 3960yd=mi d) 120in=yd

e) 9240ft=mi f) 17ft=yd

g) 26in=ft h) 3696yd=mi

i) 135in=yd j) 9504ft=mi




4. Findtheequivalentmeasurement.Whennecessary,expressyouranswerasadecimaltothenearesthundredth.

a) 8500ft=mi b) 45in=ft

c) 8.25ft=yd d) 1.4yd=in

e) 2323yd=mi f) 3.42mi=ft

g) 451 ft=in h) 19

32 yd=ft

i) 2743 in=yd j) 1.7mi=yd




5. Writetheequivalentvalue.

Inches Feet Yards Miles

1

5632

4752

44352

2.4

2816

8712

85536

6. ThedeepestpartofouroceansistheMarianaTrench 7. Themarathonraceis26miles,385yardsinlength. at11947yardsdeep.Howmanymilesdeepisthe Howmanyfeetlongisthemarathon? trench?




8. Arectangularwindowis5feet3inchesinlength, 9. JanZeleznyoftheCzechRepublicsettheworld and3feet6inchesinwidth.Iftheperimeterofthe recordinthejavelinthrowat107yards,2feet, windowmusthavecaulkingthatcosts32¢perinch, 11inchesonMay25,1996.Howmanyincheswas howmuchwillthecaulkingcost? histhrow?

10. Thefloorofahousemeasures12feet,9inchesby 11. Aswimmingpoolis50ftlong,22ftwide,and 17feet,4inches.Whatwillitcosttocarpetthe averages6ftdeep.Howmanycubicyardsof floorifcarpetcosts$27.00persquareyard? waterwillithold? (Assumenowaste.)


Section 1.3 - Converting Metric and Imperial Systems


Ifyoustudyfieldssuchasengineering,chemistryornursing,youwillneedtoconvertmeasurementsbetween thetwosystems.Toconvertbetweenmetricandimperialmeasures,itishelpfultohaveequivalentvalues. Herearesomecommonconversionfactors.

MetrictoImperial ImperialtoMetric1kilometre≈0.621miles1metre≈1.094yards1metre≈3.280feet1centimetre≈0.394inches

1mile≈1.609kilometres1yard≈0.914metres1foot≈0.305metres1inch=2.54centimetres

Comparison Size

(Scale10:1)

Ayardisslightlysmallerthanametre.

(Scale160934:1)

Akilometreisslightlymorethanhalfamile.

(Scale1:1)

Aninchismorethantwiceacentimetre.

1mile

1kilometre

1inch

1centimetre

1metre

1yard

Converting Metric and Imperial Systems1.3




Findthemissinglengths.

a) 6m=yds

b) 6yds=m

c) 3km=ft

d) 4mi=cm

e) 25cm=mi

►Solution: a) 1m=1.094yds

then6m=6#1m=6#1.094yds=6.564yds

b) 1yd=0.914m

then6yds=6#1yd=6#0.914m=5.484m

c) 1m=3.280ft

then3km=3000m=3000#1m=3000#3.280ft=9840ft

d) 1mi=1.609km,1km=1000m,1m=100cm

then4mi=4#1mi=4#1.609km=6.436km

and6.436km=6.436#1km=6.436#1000m=6436m

and6436m=6436#1m=6436#100cm=643600cm

e) 1cm=0.01m,1m=0.001km,1km=0.621mi

then25cm=25#1cm=25#0.01m=0.25m

and0.25m=0.25#1m=0.25#0.001km=0.00025km

and0.00025km=0.00025#1km=0.00025#0.621mi=0.000155mi

Example 1




1.3 Exercise Set

1. ConvertfromMetrictoImperial.(Answerstothreedecimalplaces.)

a) 1km=mi b) 1m=yd

c) 1m=ft d) 1cm=in

e) 0.621km=mi f) 27.3m=yd

g) 19.8m=ft h) 47.9cm=in

i) 3.7km=yd j) 63m=in




2. ConvertfromImperialtoMetric.(Answerstothreedecimalplaces.)

a) 1mi=km b) 1yd=km

c) 1ft=m d) 1in=cm

e) 0.621mi=km f) 27.3yd=m

g) 19.8ft=m h) 47.9in=cm

i) 3.7yd=km j) 63in=m




3. Findtheequivalentmeasurement.(Answerstothreedecimalplaces.)

a) 110yds=m (ThelengthofaCanadianfootballfield.)

b) 30cm=in (Thelengthofaruler.)

c) 533km=mi (ThedistancefromEdmontontoSaskatoon.)

d) 6ft,1in=m (Theheightofthisauthor.)

e) 70mi/hr=km/hr (ThespeedlimitontheInterstateHighwayintheUnitedStates.)

f) 110km/hr=mi/hr (ThespeedlimitonsectionsoftheTransCanadaHighway.)

g) 1253mi=km (ThedistancefromVancouvertoLosAngeles.)




3. h) 50km/hr=mi/hr (Thespeedlimitonmostcitystreets.)

i) 6ft,3in=m (TheheightofbasketballplayerSteveNash.)

j) 6.5mm=in (Thethicknessofapencil.)

k) 1815.4ft=km (TheheightoftheCNtowerinToronto.)

l) 94.51millionmiles=millionkm (ThedistancefromtheEarthtotheSun.)

m) 261millionkm=millionmiles (ThemaximumdistancebetweenEarthandVenus.)




4. Completethetable.

Object mm ft yd cm in mThicknessofhardwoodfloor 19

Heightofaroom 9

Widthofafootballfield 55

Lengthofapencil 18

Heightofatable 29

Ahomeruninbaseball 135




5. Findthemissinglengths.

a) 25squaremetres=squareyards b) 40squarefeet=squaremetres

c) 75squareinches=squarecentimetres d) 5squaremetres=squareinches

e) 8cubicmetres=cubicyards f) 75cubicfeet=cubicmetres

g) 40cubicinches=cubiccentimetres h) 250cubiccentimetres=cubicinches




6. ThespeedlimitonB.C.highwaysis110km/hr. 7. Ascalemodelofthefishingschooner,theBluenose, Johnistravellingat70mi/hr.IsJohnspeeding? is20cmlong.Every3centimetresonthemodel represents8yardsontheBluenose.Howmany metreslongistheBluenose?

8. Amaphasascaleshowing2inchesrepresents 9. Oneofthemostanticipatedrecordsintrackand 9kilometres.Howmanymilesapartarecities fieldwasbreakingthe4minutemile.Ifarunner thatare11.3inchesapartonthemap? ranthe1500metreraceatthesameaveragespeed, whattimewouldhetake,tothenearestsecond?

10. Countingtheendzones,aCanadianfootballfield 11. Ametrictonisthemassof1m³ofwater.How measures160yards#55yards.Howmanysquare manycubicfeetis1m³? metresisthisfield?




12. Mikeisrunningat7.5milesperhour.Howmany 13.Marcusisskatingat20m/s.Whatishisspeedin metrespersecondisherunningat? mi/hr?

14. Theknotisaunitofspeedequaltoonenautical 15.Oneknotishowmanym/s? mileperhour,approximately1.151mi/hr.What isaknotinkm/hr?


Section 1.4 - Surface Area and Volume of Prisms


Prismsarenamedaccordingtotheshapeoftheirbases.

Thefourprismsshownarerightprismsbecausethelateral(vertical)edgesareperpendiculartotheplaneof thebaseandthetop.

Ifthebaseisaregularpolygon(allsidesequal)thenthecorrespondingprismiscalledaregularprism.

Surface Area of Prisms

Thesurfaceareaofprismisthesumoftheareasofallthefaces,orsurfaces,thatenclosetheprism.

Cube

SurfaceArea= l l l6 6 2# # =^ h

RectangularPrism

SurfaceArea= lw lh wh2 2 2+ +

TriangularPrism

SurfaceArea=2 ab ac bc cd21

+ + +` j

= ab ac bc cd+ + +

whered a b2 2= +

Surface Area and Volume of Prisms1.4

Cube RectangularPrism HexagonalPrismTriangularPrism

l l

l

l w

h

♦ 21

a b

cd




Findthesurfacearea.

a)

b)

c)

►Solution: a) SurfaceArea= l6 2#

=6 3 54 cm2 2# =

b) SurfaceArea= lw lh wh2 2 2+ +

=2 6 3 2 6 4 2 4 3# # # # # #+ +

=36 48 24+ +

=108 cm2

c) 5 12x 1692 2 2= + =

x 169 13= =

SurfaceArea=ab ac bc cd+ + +

=5 12 5 4 12 4 13 4# # # #+ + +

=60 20 48 52+ + +

=180 in2

Note: When the units of length are: mm, cm, m, etc., the units of area are: mm², cm², m², etc.

Example 1

3cm 3cm

3cm

3cm

4cm

6cm

5in 12in

4inxin




Volume of Prisms

Volumeisthemeasureoftheamountofspacecontainedinasolid.

Whiletheformulausedforfindingthesurfaceareaofaprismdiffersdependingontheshapeoftheprism,the formulausedtocomputethevolumeofaprismisthesameforallshapesofprisms.

Volume of a Prism

IfBistheareaofthebaseoftheprism,andhistheheightoftheprism,then:Volume=B h#

Cube

Areaofbase=l l l2# =

Height=l Volume=B h l l l2 3# #= =

RectangularPrism

Areaofbase=l w# Height=h Volume=B h l w h# # #=

TriangularPrism

Areaofbase= ab21

Height=h

Volume=B h ab h21

# #=

TrapezoidPrism

Areaofbase= a b c21+^ h

Height=h

Volume=B h a b ch21

# = +^ h

l l

l

l w

h

a b

hd

a

b h

c




Findthevolume.

a) b)

c) d)

►Solution: a) Volume=l3

=23

=8 m3

b) Volume=l w h# #

=10 5 3# #

=150 ft3

c) Volume= ab h21#

=21 3 2 4# # #

=12 mm3

d) Volume=Areaofatrapezoid h#

=213 5 10 12$ #+^ h

=270 m3

Note: When the units of length are: mm, cm, m, etc., the units of volume are: mm³, cm³, m³, etc.

Example 2

2m 2m

2m

4mm

3mm

2mm

5ft

3ft

10ft

10m12m

5m

3m




1.4 Exercise Set

1. Determinethesurfaceareaandvolumeoftherightprism.

a) b)

S.A.= S.A.=

V = V =

c) d)

S.A.= S.A.=

V = V =

5ft 5ft

5ft

5cm

4cm

10cm

8mm

6mm10mm

3mm

20m 9m5m

13m12m

10m

15m




1. e) f)

S.A.= S.A.=

V = V =

g) h)

S.A.= S.A.=

V = V =

10yds20yds

7yds

20ft30ft

8ft

6ft

6cm

8cm

8m12m




1. i) j)

S.A.= S.A.=

V = V =

k) l)

S.A.= S.A.=

V = V =

8m 12m

5m

4m

3m

8m

9m7m

7m

4mm

3mm5mm

2mm

5ft

6ft12ft

4ft




2. AnOlympicsizeswimmingpoolis50mlong,21m 3. Thedimensionsfora wideand2mdeep.Howmanylitresofwaterare barnareshown.What neededtofillthepool?(1m³=1000l) isitsvolume?

4. Abedroomhaslength4.8m,width4.2mand 5. Ifastackof500sheetsofpapermeasures height2.2m. 21.59cm#27.94cm#5cm

a) Findthesurfaceareaofthewallsandceiling. a) Whatisthethicknessofeachsheetofpaper?

b) Ifonelitreofpaintcovers8m²,andMary b) Whatisthevolumeofonesheetofpaper? appliestwocoatsofpaint,howmanylitresof paintwillsheneedforthisroom?

10m3m

10m6m

2m




6. Thedimensionsfora 7. Asectionoffreeway1kmlongand15mwideisto sandboxareshown. beresurfacedwith25cmofasphalt.Whatisthe Ifthewallsofthe volumeofasphaltneeded,incubicmetres? sandboxaretobe paintedinsideandoutside, whatisthetotalareatobe painted?

8. Thelengthofarectangularsolidisthreetimesthe 9. Apieceofcheeseisin widthandtheheightistwicethewidth.Ifthe theshapeofaright volumeis162cm³,whatisthewidthofthe triangularprismofthe rectangularsolid? givendimensions. Whatisthevolumeof thepieceofcheese?

3m2.5m

0.6m

6cm 8cm

5cm




10. Acubehasavolumeof27cm³.Whatisits 11. Trishaboughta40litrefishtank.Shewastoldthat surfacearea? afishtankcouldsupportonegoldfishforevery 2500cm³ofwater.Howmanygoldfishcanthe tanksupport?(1litre=1000cm³)

12. Arockissubmergedinarectangularprismwhose 13. Thebaseofarighttriangularprismhassidesof basemeasures10cmby15cm.Ifthewaterlevel 3,4,5cm.Thevolumeoftheprismis84cm³. risesfromaheightof8cmto11cm,whatisthe Findtheheightoftheprism. volumeoftherock?




14. Thedimensionsfora 15. Findthe swimmingpoolin volumeof theshapeofa thebarn. prismareshown. Howmanylitresof waterdoesittaketofill thepool?(1000litres=1m³)

16. Adrainagecanalisaright 17. Thevolumeofanopen-toppedboxwitha trapezoidprismofthe squarebaseis32cm³.Ifitsheightis2cm, indicateddimensions. whatisthesurfaceareaofthebox? Whatisthevolume oftheprism? 13m

10m

50m

20m

16 m

32 m 6 m4 m6m

5m

4m

10m

1m




18. Aregularhexagonalbasedprismhassidesof4cm 19. Arightprismwhosebaseisarhombushas andaheightof5cm. diagonalsof12cmand16cm,withaheightof6cm.

a) Whatisthesurfaceareaoftheprism? a) Whatisthesurfaceareaoftheprism?

b) Whatisthevolumeoftheprism? b) Whatisthevolumeoftheprism?

20. Findthesurfacearea 21. Findthevolumeofarighttriangularprismwith oftheprism. height x2 1+^ h.Thebaseisanisoscelesright trianglewithahypotenuseof x2 .x

x

2x

2x

4x


Section 1.5 - Surface Area and Volume of Pyramids


Surface Area of a Pyramid

Apyramidisasolidfigure,whoselateralfacesaretrianglesthathaveonepointincommon.

Thisiscalledasquarepyramidbecauseitsbaseissquare.Determiningthesurfaceareaofthispyramid requiresfindingtheareaofthebase,plustheareaofthefourlateralfaces.

Determinethesurfaceareaofthesquarepyramid.

►Solution: Ifthedistancefromthecentreofthebasetotheedgeis5,thenthelengthandwidthofthe baseis10.

Areaofbase=10 10 100# = cm²

Tofindtheslantheight,usethePythagoraeantheorem.

5 12s 1692 2 2= + =

s 13= cm

Areaofonelateralface=21 10 13 65# =^ h cm²

Thereforethesurfacearea= ( )100 4 65 360+ = cm²

Example 1

Surface Area and Volume of Pyramids1.5

PerpendicularHeight(h)

SlantHeight(s)

LateralEdge

Base(B)

LateralFace

SlantHeight(s)

5cm

10cm

12cm




Volume of a Pyramid

Consideracubeoflength2x.Drawthefourdiagonalsofthecube.Theareaofthebaseofeachpyramidis x x x2 2 2 2=^ ^ ^h h h .Thepyramidswillmeetatthecentreofthecubewithheightxfromthebase.Ifyoulook carefully,youwillseesixequalsquarebasedpyramids,onefromeachofthesixsidesofthecube.

LetthevolumeofeachsquarebasepyramidbeV.Thenthevolumeofthecubeis6V.

So V x6 2 3= ^ h ,thereforeV x x x x x

61 2

61 2 2

31 23 2 2

= = =^ ^ ^ ^ ^h h h h h

WecansaythevolumeofeachpyramidisV=31 #(areaofbase)#(height).Thisformulaistrueforall

pyramids.

Volume of a Pyramid

ThevolumeofanypyramidwithareaofbaseB,andheighthisV B h31#= ^ h

Calculatethevolumeoftherectangularpyramid.

►Solution: V B h31#= ^ h

31 6 8 10#= ^ ^h h

160 in3=

Example 2

2x

x

2x

2x

8in6in

10in




Usingthenet,determinethevolumeofthesquarebasedpyramid.

►Solution: Solvefortheheightofpyramid,h.

h 5 132 2 2+ =

169 25h 1442= - =

h 12= cm

V B h31#= ^ h

31 10 10 12#= ^ ^h h

400 cm3=

Thevolumeofarectangularpyramidis54cm³.Determinethedimensionsofitsbase,ifthe lengthofthebaseistwicethewidth,andtheheightisthreetimesthewidth.

►Solution: V = B h31 #^ h

54= x x x31 2 3#^ ^h h

54= x2 3

27= x3

x = 3

Thereforethebaseis3cm# 6cm.

5cm

13cm h

Example 3

Example 4

10cm

13cm




1.5 Exercise Set

1. Findthetotallateralarea(notincludingthebase)ofeachregularpyramid.

a) Triangularbasewithsides5cmandslantheight b) Squarebasewithsides5cmandslantheight8cm. 8cm.

c) Pentagonalbasewithsides5cmandslantheight d) Hexagonalbasewithsides5cmandslantheight 8cm. 8cm.

e) Octagonalbasewithsides2cmandslantheight f) Decagonalbase(10sides)withsides2cmand 8cm. slantheight8cm.

g) Dodecagonalbase(12sides)withsides2cm h) Hexadecagonalbase(16sides)withsides2cm andslantheight8cm. andslantheight8cm.




2. Findthevolumeofeachpyramid.

a) Areaofbaseis15m²andheightis5m. b) Areaofbaseis18.6in²andheightis7in.

c) Areaofbaseis270ft²andheightis8yds. d) Areaofbaseis36m²andheightis200cm.

e) Areaofbaseis97.2cm²andheightis86mm. f) Areaofbaseis1200in²andheightis 43 ofayard.

g) Areaofbaseis3m²andheightis340mm. h) Areaofbaseis9m²andheightis2ft.




3. Findthesurfaceareaandvolume.

a) b)

S.A.= S.A.=

V = V =

c) d)

S.A.= S.A.=

V = V =

6cm6cm

4cm

10cm10cm

12cm

16m16m

17m 7m

Alledgesare10m




4. Whatistheratioofthe 5. Findthesurfaceareaof volumeofthepyramid pyramidABCDEinscribed tothevolumeofthe insideacube. rectangularsolid?

6. Twopyramidsareinscribedincubes. 7. Aregularhexagonalpyramidhasbaseedgesof length1cm,andlateraledgesoflength2cm. i) ii)

a) Whichpyramidhasthegreatestvolume? a) Whatisthevolumeofthepyramid?

b) Whichpyramidhasthegreatestsurfacearea? b) Whatisthesurfaceareaofthepyramid?

A

B

E

C

D




8. Arightrectangularpyramidhasabase6cmby 9. Findthelateralsurfaceareaofasquarepyramid 8cm,andaheightof10cm.Whatisthevolume whoseheightis8cmandwhosebaseonedgeis ofthepyramid? 12cm.

10. Thebaseofaregulartriangularpyramidisan 11. Astonemonumentistheshapeofapyramid. equilateraltriangle.Thevolumeofthepyramid Theareaofthebaseis81m²andtheheightofthe is27cm³andtheheightis3 3 cm.Whatisthe monumentis10m.Thestoneweighsabout lengthofthesideofthebase? 3000kg/m³.Howmanytonnesdoesthemonument weigh?(2000kg=1metrictonne)




12. Findthesurfaceareaandvolumeofaregular 13.Analtitudeofasquarepyramidis10cm,andthe hexagonalpyramidwithbaseedge14inches sidesofthebaseare15cm.Findtheareaofa andheight24inches. crosssectionatadistanceof4cmfromthetop ofthepyramid.

14. Ifyoudoubleallthedimensionsofapyramid,what 15. Apyramidhasarighttriangularbaseanda doesitdotothesurfacearea?Volume? volumeof120m³.Theshortersidesofthebase are6mand9mlong.Findtheheightofthe pyramid.




16. Findthetotalsurfaceareaofaregularpyramid 17. Apyramidhasarectangularbasewithlengthtwice whoseheightis15cmandwhosebaseisasquare thewidth.Iftheheightis6cmandthevolumeis withsides16cm. 256cm³,whatarethedimensionsoftherectangular base?

18. Findthetotalsurfaceareaofaregularpyramid 19. Ifthediagonalsofacubearedrawn,theymeetat whoseheightis3cm,andwhosebaseisarectangle thecentreofthecube.Whenthediagonalsare withsides6cmand8cm. drawn,howmanypyramidsareformed?Ifthe edgeofthecubeis9cm,findthevolumeofone ofthepyramids.


Section 1.6 - Surface Area and Volume of Cylinders, Cones, and Spheres


Cylinders

Acylindercanbethoughtofasarectangularprismwithacircularbase.Forthatreason,calculatingthesurface areaandvolumeisverysimilartohowitisdoneforaprism.

SurfaceArea= Areaoftopandbottomfaces + Areaoflateralside = 2 r2r + rh2r

Volume = Areaofbase # height = r2r # h

Surface Area and Volume of a Cylinder

SurfaceArea= r rh2 22r r+

Volume= r h2r

Determinethesurfaceareaandvolumeofacylinderwithradius4cmandheight6cm.

►Solution: SurfaceArea= r rh2 22r r+

=2 4 2 4 62# # #r r+

=80 251.3 cmcm2 2-r

Volume = r h2r = 4 62# #r

= cm . cm96 301 63 3-r

Note: The answer “80 cm2r ” is accurate, “251.3 cm2” is rounded to one decimal place.

r

h

r

h

Example 1

Surface Area and Volume of Cylinders, Cones, and Spheres1.6




Cones

Aconecanbethoughtofasapyramidwithacircularbase.Forthatreason,calculatingthesurfaceareaand volumeisagainverysimilartohowitisdoneforapyramid.

(l=slantheight,withl r h2 2 2= + )

SurfaceArea = Areaoflateralside + Areaofbase = rlr + r2r

Volume=31 # Areaofbase # height

=31 # r2r # h

Surface Area and Volume of a Cone

SurfaceArea= rl r2r r+ ,withl r h2 2 2= +

Volume= r h31 2r

Determinethesurfaceareaandvolumeofaconewithradius5ftandheight12ft.

►Solution: Ifr=5ftandh=12ft,thenl r h l5 12 169 132 2 2 2 2"= + = + = = ft

SurfaceArea= rl r2r r+

= 5 13 52# # #r r+

= 90 ft2r

Volume = r h2r = 5 122# #r

= 300 ft3r

Example 2

LateralSide Baser

h r

l

l+




Spheres

Aspheremaybethoughtofasallpointsinspacethatareafixed,orequaldistancefromagivenpoint.The givenpointisitscentre,andthedistance(r)istheradiusofthesphere.Halfofasphereiscalledahemisphere.

Toprovetheformulasforthesurfaceareaandvolumeofasphererequirestheuseofcalculus.Soherewe simplygivethemtoyou.

Surface Area and Volume of a Sphere

SurfaceArea= r4 2r

Volume= r34 3r

Determinethesurfaceareaandvolumeofaspherewithradius6mm.

►Solution: SurfaceArea= r4 2r

=4 62# #r =144 mm 2r

Volume= r34 3r

=34 63# #r

= mm288 3r

Determinethesurfaceareaandvolumeofahemispherewithcircumference8r cm.

►Solution: C r r2 8 4"r r= = =

SurfaceArea r r2 2 2r r= + Volume r332 r=

r3 2r= 2 3

32 r= ^ h

3 4 2r= ^ h 316 r= cm³

48r= cm²

Example 3

r

Example 4




1.6 Exercise Set

1. Determinethesurfaceareaandvolume.

a) b)

S.A.= S.A.=

V = V =

c) d)

S.A.= S.A.=

V = V =

4m

8m6mm

10mm

1.5mm

3cm

2cm

4cm

3cm

5m3m

8m




1. e) f)

S.A.= S.A.=

V = V =

g) h)

S.A.= S.A.=

V = V =

8cm

3cm

3cm

12cm

4cm

9cm

4cm9cm

5m

18m




1. i) j)

S.A.= S.A.=

V = V =

2. Agreenhouseissemi- 3. Soupissoldintwocansizes.Thecanis cylindricalinshape.If cylindricalinshape.TheheightofcanAistwice clearvinylisusedto theheightofcanB.TheradiusofcanBistwice coverthegreenhouse, theradiusofcanA.CanBcoststwiceasmuch howmuchmaterialis ascanA.Whichcanisthebetterbuy? neededif10%extrais requiredforwaste?

5cm 10ft

4m10m




4. Acertaintypeofdrainagetileismadefromclay. 5. IfrectangleABCDis Eachtileisacylindricalshell30cmlongwithan isrevolvedaboutthe insideandoutsidediameterof10cmand12cm. lineAD,whatisthe Whatisthevolumeofeachclaytile? volumeofthespace throughwhichitmoves?

6. Asteelpipehasaninsideradiusof4cmand 7. Apapercupistheshapeofaconewithradius3cm anoutsideradiusof5cm.Calculatethevolume andheight4cm.Howmuchpaperisneededto ofsteelina2.5mlengthofpipe. makethiscup?

2m

4mA

B

D

C




8. Acylinderhasavolumeof320πcm³anda 9. Supposetheradiusandheightofacylinderare heightof5cm.Findtheradiusofthecylinder. bothdoubled.Byhowmuchisthesurfacearea increased?Byhowmuchisthevolumeincreased?

10. Acylinderofradius6cmandheight18cmisfull 11. Calculatethevolumeofthe ofwater.Whatmustbetheheightofaconewith solidformedbyrotatingthe radius9cmtoholdthesameamountofwater? figurearoundthelineAB.

2cm

3cm

A

B

C




12. Twowaterpipesofthesamelengthhavearadius 13. Calculatetheheightofarightcircularconewhose of3cmand4cm.Ifthesetwopipesarereplaced volumeis75π,andwhosebasehasadiameterof byonepipeofthesamelength,whatmustbethe 10ft. radiusofthenewpipeifthevolumeisthesame?

14. Whatisthelateralsurfaceareaofarightcircular 15. Thediameterandheightofarightcircularconeare coneofheight12inandradius5in? thesame.Ifthevolumeoftheconeis144πcm³, whatistheradiusofthecone?




16. Iftheradiusofarightcircularcylinderis3cm, 17. Acylindricalblockofcheese andthetotalsurfaceareais42πcm²,whatisthe hasa60°sliceremoved.Ifthe height? volumeoftheremovedsliceis 18πcm³,whatistheheightof theblockofcheese?

18. Arightcylinderhasheight8mmand 19. Acanofpeashasaheightof15cmanda circumference6πmm.Whatisthevolume circumferenceof10πcm.Whatistheamountof ofthecylinder? paperneededforthelabelonthecanofpeas?

6cm




20. Thetotalsurfaceareaofarightcircularconeis 21. Asphericalmetalshellis1cmthick.Iftheoutside 90πcm².Itsradiusis5cm.Whatisthevolume diameteris8cm,findthevolumeofmetalinthe ofthecone? shell.

22. Asiloiscomposedofa 23. Ahemispherehasa hemisphereontopofa diameterof10cm. cylinder.Ifthecylinder Whatisthetotal heightanddiameterare surfaceareaofthe both8m,whatisthe hemisphere? volumeofthesilo?

10cm8m

8m




24. Ahemisphericalroom 25. Ifasphereandaconehavethesameradiusand requiresfourcansof volume,whatmustbetheheightoftheconein painttopaintthefloor. termsoftheradius? Howmanycansof paintarerequiredto paintthewalls?

26. Aspherewitharadiusof4cmhasa45°section 27. Aspherewitharadiusof4cmhasa45°section removed.Whatistheremainingvolumeofthe removed.Whatistheremainingsurfaceareaof sphere? thesphere?




28. Aspherefitsexactlyinto 29. Asphereofradiusrisinscribed acube.Whatistheratio inaconewithdiameter ofthesurfaceareaofthe AC=AB=BC.Findthe spheretothesurfacearea ratioofthevolumeofthe ofthecube? spheretothevolumeof thecone.

30. Threetennisballsarepackedtightlyindifferentcontainers. Whatistheratioofthevolumeofthetennisballstothe volumeoftheboxinfiguresa,bandc?

a) b) c)

A

B

C

r


Section 1.7 - Chapter Review


Section 1.1

1. Add.

a) 4.6m+25cm+240mm=mm=cm=m

b) 1.2km+612m62583cm=km=m=cm

c) 0.0075km+0.32m+61.3cm=km=m=cm

d) 0.25km+6200cm+12000mm=m=cm=mm

Chapter Review1.7




Section 1.2

2. Add.

a) 0.5mi+438yd+812ft=mi=yd=ft

b) 720in+96ft+51yd=in=ft=yd

c) 0.2mi+32yd+78ft=mi=yd=ft

d) 0.25mi+360yd+6600in=mi=yd=ft




Section 1.3

3. ConvertfromMetrictoImperialunits.(Answerstothreedecimalplaces.)

a) 5.3km=mi b) 63cm=in

c) 345mm=in d) 82m=ft

e) 0.28km=yd f) 74m=yd

g) 436cm=ft h) 428000mm=mi




4. ConvertfromImperialtoMetricunits.(Answerstothreedecimalplaces.)

a) 5.3mi=km b) 63in=cm

c) 345in=mm d) 82ft=m

e) 280yd=km f) 74yd=m

g) 4.36ft=cm h) 0.0428mi=mm




Section 1.4

5. Determinethesurfaceareaandvolumeoftherightprisms.

a) b)

S.A.= S.A.=

V = V =

c) d)

S.A.= S.A.=

V = V =

12cm

9cm15cm

4cm 6in9in

8m 12m5m

3m

5ft

7ft




Section 1.5

6. Determinethesurfaceareaandvolumeofthepyramids.

a) b)

S.A.= S.A.=

V = V =

12in12in

8in

14cm14cm

24cm




7. Apyramidandaprismbothhavethesamebaseand 8. Apyramidhasarighttriangularbasewithsides height.Determinetheratioofthevolumes. 5cm,12cmand13cmlong.Determinetheheight ofthepyramidifthevolumeis120cm³.

9. Twopyramidsaresimilar.Thevolumeofthelarger 10. Twopyramidsaresimilar.Thevolumeofthelarger pyramidis500cm²andtheratioofthebaseareasof pyramidis500cm²andtheratioofthebaseareasof thepyramidsis9to25.Whatistheratioofthe thepyramidsis9to25.Whatisthevolumeofthe heightsofthepyramids? ofthesmallerpyramid?




Section 1.6

11. Determinethesurfaceareaandvolumeofthesolids.

a) b)

S.A.= S.A.=

V = V =

c) d)

S.A.= S.A.=

V = V =

4ft

6in

12in

8in

9in

6mm10mm

♦ 56

2cm

1cm

3cm

2cm


1.1 Metric Systems - Miss Johnston's Math...

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