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