MPM1D–Unit6:Measurement–Lesson1 Date:______________Learninggoal:howtousePythagoreanTheoremtofindunknownsidelengthinarightangletriangle.
PythagoreanTheorem
Investigate:1.Whattypeoftriangleisinthecentreofthepicture?2.Completethetablebelow
Sidea= AreaofA=
Sideb= AreaofB=
Sidec= AreaofC=3.HowistheareaofAandtheareaBrelatedtotheareaofC?Example1 a)Labelthehypotenuse a)Labelthehypotenuse b)Calculatethemissingsidelength b)Calculatethemissingsidelength
10 cm
11 cm
9 mm 7 mm
PythagoreanTheorem
Example2:Atelevisionisdescribedasa20”televisionifthescreenhasadiagonallengthof20”.Ifthescreenofa20”flat–screentelevisionhasaheightof12”,whatisthewidth?Example3:Aladderthatis8.5mlongleansagainstawall.Thefootoftheladderis2.3mfromthebaseofthewall.Howfarupthewalldoestheladderreach? Example4:Therangerhastosearchtheareaoftheoutlinedinthetriangle.Determinetheperimeterandareathattherangermustcover.
BeachCabin
3km 2.4km
RangerStation
Example5:Determinetheheightofthecone.
Assignment6.1:PythagoreanTheorem**Roundallanswerstotwodecimalplaces**
1. Solveforthemissingsideinthefollowingtriangle:
2. A wall is supported by a brace 10 feet long, as shown in the diagrambelow.Ifoneendofthebraceisplaced6feetfromthebaseofthewall,howmanyfeetupthewalldoesthebracereach?
3. WhenArnoldswimslapsinhisrectangularswimmingpool,heswimsalongthediagonalsohedoesn’thavetoturnaroundsooften.FindthedistanceArnoldtravelsbyswimmingoncealongthediagonal.
4. Thethreesidesofatriangleare18cm,24cmand30cm.Determinewhetherthetriangleisarighttriangle.Justifyyouranswersusingcalculations.
5. Determinethevalueofxandy.
6. TownsA,B,C,anddaresituatedasshownonthediagram. a.HowfarisitfromtownBtotownD? b.HowfarisitfromtownBtotownC? 6.1Answers1.502.8ft.3.10.31m4.Itisarightangletriangle.Thesumofthesmallesttwosidessquaredequalsthelargestsidesquared.5.x=25,y=206.a)7.14kmb)9.33km
3.2m
9.8m
40 cm
30 cm
7Km
6Km
10Km
B
A C
D
MPM1D–Unit6:Measurement–Lesson2 Date:______________Learninggoal:howtocalculatetheareaandperimeterforcompositeshapes.
CompositeShapes
Example1:Calculatetheperimeterandareaofthefollowingfigure.
Example2:Thefiguretotherightismadeupofatriangleandarectangle.Determinetheareaofthefigure.Example3:Calculatetheareaoftheshadedregion.
Example4:Determineasimplifiedexpressionfortheperimeterandareaofthefollowingfigures.
a)
b)
Example5:Thecircumferenceofacircleis43.96feet.Whatisthediameterofthecircle? Example6:Theperimeterofarectangleis34cm.Ifthelengthoftherectangleis11cmthenwhatisthewidthoftherectangle? Example7:Theareaofasquareequalstheareaofacircle.Ifthesquarehasasidelengthof8cmdeterminetheradiusofthecircle.
Assignment6.2:CompositeShapes**Roundallanswerstothenearesthundredth**
1. Usethefigureontherighttoanswerthequestions.
a)Whatistheareaofthesquare?b)Whatistheareaofthetriangleontheleft?c)Whatistheareaofthecompositefigure?
2. Findtheareaofthefollowingshadedregions.
a)
b)
3. Determineasimplifiedexpressionfortheperimeterandareaofthefiguretotheright.
4. Afigureisatriangle
composedofandasquare,asshownbelow.Theareaofthesquareis144m2.Whatistheheightofthetriangle?Showyourwork.
5. Thefigurebelowismadetwosemicircles.Whatistheareaoftheshadedregion?Showyourwork.
6. Acircleandatriangleofanequalarea.Ifthecirclehasaradiusof4cmandthetrianglehasaheightof15cm,determinethebaseofthetriangle.
7. Theareaofatriangleis32cm2.Iftheheightofthetriangleis7inchesthenwhatisthebaseofthetriangle?
8. Thebaseofaparallelogramis17mm.Iftheareaoftheparallelogramis82mm2,whatistheheightoftheparallelogram?
9. Theareaofarectangleis42cm2.Ifthelengthoftherectangleis5cm,whatistheperimeteroftherectangle?
10. Arightangletrianglehasaheightof6inchesandhypotenuseof9inches.Whatistheareaofthetriangle?
6.2Answers1.a)25units2b)15units2c)55units22.a)7.73m2b)71.92units23.perimeter=18x,area=16units2
4.6.7m5.131.95cm26.6.7cm7.9.14cm8.4.82mm
9.26.8cm10.20.13inches2
MPM1D–Unit6:Measurement–Lesson3 Date:______________Learninggoal:howtodeterminethevolumeofprisms&cylinders.
VolumeofPrisms&Cylinders
___________________istheamountofspacea3-dimensionalobjectoccupies.3-dimensionalobjectsarecalled__________ifthefigurehasthesameshapeonboththetopandthebottom.RECTANGULARPRISMItiscalledarectangluarprismbecausethebaseisa______________________________Example:
Example1:Calculatethevolumeoftherectangularprismtotheright.
Thebaseoftheprismisthesidethatissittingontheground.Thisshapeisarectanglewithlength5cmandwidth7cm.
Aprismisa3-dimensionalobjectbecause
length
height width
12 cm
7cm
5cm
VolumeTofindthevolumeofaprism,findthebaseareaandmultiplybytheheight
𝑉 = 𝐴 × 𝐻
Becausewearemultiplying3dimensions,volumeisexpressedin“cubic”units.Forexample:cm3,orm3.
Example2:Calculatetheareaofthebase,andthencalculatethevolume.
Example3:Calculatethevolumeofthefollowingcompositeshape.Example4:Staceyisfillingherfishtankwithhercylindericalcup.Hercuphasaradiusof3.5cmandaheightof10cm.Herfishtankis30cmwide,50cmlong,and40cmtall.Howmanycupswillittaketofillherfishtank?
Assignment6.3:VolumeofPrisms&Cylinders**Roundallanswerstothenearesthundreth**
Forquestions1-4,findthevolumeoftheshapedrawn.
5. DairyFreshpackagestwocartonsofmilk.Thelargecartonis8cmby8cmby18cm.Thesmallcartonis
6cmby6cmby8cm.Howmanysmallcartonsofmilkwouldfillonelargeone?6. Agoldbaris15cmby8cmby3cmandisworth$350000.Whatisthevolumeofthebarincubic
centimetres?Howmuchisonecubiccentimetrework?7. Acylinderjustfitsinsidea10cmby10cmby10cmbox.
a) Determinethevolumeofthebox.b) Determinethevolumeofthecylinder.c) Determinethevolumeoftheemptyspaceintheboxwhenthecylinderisplacedinside.
8. Tracyplanstocreateanicesculptureforatablecentrepieceatherfriend’shousewarming.Theiceshe
willstartwithisshown.CalculatethevolumeoficeTracywillneedtomakethesculpture.
6.3Answers1.810m3 5.4cartons2.3392.92ft3 6.$972.223.565.49mm3 7a)1000cm3b)785.40cm3c)214.60cm34.300in3 8.75.71cm3
MPM1D–Unit6:Measurement–Lesson4 Date:______________Learninggoal:howtocalculatethesurfaceareaofprisms&cylinders.
SurfaceAreaofPrisms&Cylinders
Rememberthatwhena2Dnetisfoldedtogetheritturnsintoa3Dshape.Example1:Findthesurfaceareaoftherectangularprism
è
Surafaceareais______________________________________________________________.
Example2:Findthesurfaceareaofthefollowingprism.
12 cm
10 cm
20 cm
Example3:Findthesurfaceareaofthefollowingcylinder.
Example4:Findthesurfaceareaofthefigurebelow.Theuppercylinderiscenteredonthelowerone.
8 cm
2 cm
Assignment6.4:SurfaceAreaofPrisms&Cylinders**Roundallanswerstothenearesthundreth**
1. Calculatethesurfaceareaofthefollowingshapes.
2. Acanofsoupis10.3cmhighanditsdiameteris6.7cm.Howmuchpaperisneededtomakethelabel?
3. Awedgeofcheeseisintheshapeofatriangularprism.Itis6cmhighandeachtriangularfacehasasidelengthof10cmandabaseof2.9cm.
(a) Howmuchplasticwrapisneededtocoverthecheese?(b) Theplasticwrapcosts$0.01/cm2.Whatisthecosttopackagethecheese?
4. Findthesurfaceareaofthefollowingcompositeshape.
5. Aboxwithasquarebasemusthaveavolumeof200cm3.Findthedimensionsoftheboxthatminimizestheamountofmaterialused.
6.4Answers1a)216cm3 b)3091.33mm32.216.8cm33a)168.86cm3b)$1.694.426.8ft35.5.85cmx5.85cmx5.85cm
(b)
MPM1D–Unit6:Measurement–Lesson5 Date:______________Learninggoal:howtodeterminethevolumeofcones,spheres,andpyramids.
VolumeofCones,Spheres,andPyramids
SPHERES
Spheresareveryuniqueshapesandassuch,itisverydifficulttofigureoutexactlyhowtofindthevolumeofasphere.Fortunatelyforus,Archimedes,theGreekmathematicianalreadyfigureditoutalongtimeago.
Example1:Findthevolumeofthegivenfiguretotheright.
RECTANGULARPYRAMIDS
Thepyramidontherighthassimilardimensionswithrespecttoitslength,widthandheight,butitsvolumeisclearlymuchlessthanthatoftherectangularprism.Infact,thevolumeofthepyramidisexactly______thevolumeofthesimilarrectangularprism.
VolumeofSpheres
𝑉 =43𝜋𝑟!
VolumeofRectangularPyramids
𝑉 =𝑙×𝑤× ℎ
3
IMPORTANTTERMSHeight:(altitude)Slantheight:Howdoespythagorushelpushere?Example6:Findthevolumeofthepyramidtotheright.CONES
Therelationshipbetweencylindersandconesissimilartotherelatioshipbetweenprismsofpyramids.
𝑉 = 𝜋𝑟!ℎ
𝑉 =𝜋𝑟!ℎ3
Example2:Findthevolumeoftheconebelow.
16 m
11 m
16 m
Example3:Aconehasvolume263.9cm3andbaseradius6cm.Whatistheheightofthecone? Example4:Determinetheamountofemptyspaceinthesquarebox.
12.5 m
23 m
25 m
24 m
13 m
24 m
15 in.
18 in.
Assignment6.5:VolumeofCones,Spheres,andPyramids**Roundallanswerstothenearesthundreth**
1. Calculatethevolumeofeachsolid.
a) b)
c) d)
2. Calculatethevolumeofaspherewithadiameterof78cm.
3. Calculatethevolumeofthefigureshowninthediagramtotheright
4. Aspherewitharadiusof46cmiscenteredinsideaspherewitharadiusof76 cm.Whatisthevolumeofthespacebetweenthetwospheres?
5. Asculptorwantstoremovestonefromacylindricalblock3feethighandturnitintoacone.Thediameterofthebaseoftheconeandcylinderis2feet.Whatisthevolumeofthestonethatthesculptormustremove?
6.5Answers
1a)2395.83m3b)1017.88in3c)1047.39mm3 d)960m3
2.248474.85cm3 4.1431058.29m3
3.1583.36cm3 5.𝜋ft3
MPM1D–Unit6:Measurement–Lesson7 Date:______________Learninggoal:Icanapplyagebratosurfaceareaandvolume.
SurfaceAreaandVolumewithAlgebra
Example1:TheEarthhasasurfaceareaof509,904,262.8km2.
a) Whatistheearth’sdiameter?
b) Whatisthesurfaceareaofeachhemisphere? Example2:Theareaofasquareequalstheareaofacircle.Ifthesquarehasasidelengthof8cmdeterminetheradiusofthecircle.
Example3:acone-shapedpapercuphasavolumeof67cm3andadiameterof6cm.Whatistheheightofthepapercup,tothenearesttenthofacentimeter? Example4:Patriceisinchargeoftheexcavationforthefoundationofabuilding.Ifaholemustbedugthatis35mby25mby12m,howmanytripswillberequiredtoremovethedirtifatrailercancarryonly15cubicmetersofdirt?Example5:Acylindricaljuicecontainerhasadiameterof8cm.Ifthesurfaceareaofthejuicecontaineris754cm2,whatistheheightofthecontainer?
Assignment6.7:SurfaceAreaandVolumewithAlgebra
**Roundallanswerstothenearesthundreth** 1. Determetheheightofaconeifthevolumeoftheconeis56cm3andthediameteris5cm.2. Ifacanofsoupcanhold64cm3ofsoup,andtheheightofthecontaineris6cm,whatistheradiusof
thefilledcanofsoup?3. Acircleandatriangleofanequalarea.Ifthecirclehasaradiusof4cmandthetrianglehasaheight
of15cm,determinethebaseofthetriangle.4. Findtheheightofacanwithasurfaceareaof326.73cm2,iftheradiusis4cm. 5. Ifthevolumeofasquare-basedpyramidis680cm3andthesidelengthsofthesquareare16cm,
whatistheheightofthepyramid?6. Agiftisputintoarectangularboxmeasuring6x6x15inches.Ifeachwrappingpapersheetcovers
60in2,howmanysheetsmustbepurchasedtocompletelywrapthegift?6.7Answers1.8.56cm2.1.84cm3.6.7cm4.9cm5.7.97cm6.8sheets