VolumeofCylinders,Cones,andSpheres
Goal:Toprovideopportunitiesforstudentstopracticecalculatingthevolumeofcylinders,
cones,andspheres
CommonCoreStandards
Geometry
Solvereal-worldandmathematicalproblemsinvolvingvolumeofcylinders,cones,andspheres.
8.G.9.Knowtheformulasforthevolumesofcones,cylinders,andspheresandusethemto
solvereal-worldandmathematicalproblems.
StudentActivitiesOverviewandAnswerKey
Station1
Studentsconstructacylinderoutofasheetofpaper.Theyexplorewhetherthewaytheychoosetorollthepaperaffectsthevolume.Theyexperimentandusecalculationstoanswerthisquestion.
Answers:Answerswillvary;thecylinderwiththe11-inchcircumference;76.9cubicinches;63.0cubicinches;thecylinderwiththe11-inchcircumference
Station2
Atthisstation,studentsusetheformulaforthevolumeofacone(V=πr2h)todeterminehow
manyicecreamconescouldfitintoacooler.Theyexplainhowtheydothis,andareinstructedtobe
carefulwiththeirunits.
Answers:πr2;4.19cubicinches;12cubicfeet;4,948cones;youneedtochangefeettoinchesorinchestofeetsoyouareworkinginthesameunit,thenyoucandividethevolumeofaconeintothevolumeofthecooler
Station3
Studentsdeterminethevolumeofseveralsphericalobjectsbyusingtheformulaforthevolumeofa
4πr3
sphere . Studentsmeasurethecircumferenceofeachobjectusingmeasuringtape,then
derivethevalueoftheradiusrusingtheformulaforthecircumferenceofacircle(C=2πr).They
thensubstitutethevalueforrintotheformulaforvolume.
Answers:Answerswillvarydependingontheobjectschosen.
Instruction
Station4
Studentsusetheformulaforthevolumeofasphere tosolveproblemsasagroup.
Answers:Answersmayvarydependingonwhetherstudentsuseacalculatorwithaπbuttonorif theyusetheapproximatevalue3.14forπ.Answersderivedusingπbutton:
1.448πcm3or1407.43cm3
2.576πin3or1809.56in3
3.16,848πyd3or52,929.56yd3
4.14,580πmm3or45804.42mm3
5.103.67cm3
6.804.25mm3
7.1357.17in3
8.3015.93 yd3
9.3053.63ft3
10.2144.66mm3
11.268.08cm3
12.113.1yd3
twosheetsofpaper;tape;minimarshmallows
calculator
foursphericalobjectsofvaryingsizes(Ping-Pongballs,orange,basketball,globe,etc.);measuringtape;calculator
calculator
Discussion Guide
Tosupportstudentsinreflectingontheactivitiesandtogathersomeformativeinformationaboutstudentlearning,usethefollowingpromptstofacilitateaclassdiscussionto“debrief”thestationactivities.
Prompts/Questions
1.Whydoesthewayyourollarectangleaffectthevolumeofthecylinderyoucreate?
2.Usingareal-lifeexample,whenisitimportanttoknowthevolumeofanobject?
3.Whatistheformulaforthevolumeofacylinder?
4.Whatistheformulaforthevolumeofacone?
5.Whatistheformulaforthevolumeofasphere?
Think,Pair,Share
Havestudentsjotdowntheirownresponsestoquestions,thendiscusswithapartner(whowasnotintheirstationgroup),andthendiscussasawholeclass.
SuggestedAppropriateResponses
Ifyourollitoneway,thelengthoftherectangleistheheightofthecylinder;ifyourollittheotherway,thelengthoftherectangleisthecircumferenceofthecylinder.
Examplesincludeacanofsoup,astoragecontainer,etc.
πr2h
πr2h)
4πr3
3
PossibleMisunderstandings/Mistakes
•Incorrectlymeasuringobjects
•Incorrectlysubstitutingmeasurementsintheformulas
•Mistakingcircumferenceforradius
•Makingarithmeticalerrors
•Notunderstandingthattheradiusishalfthediameterofanobject
VolumeofCylinders,Cones,andSpheres
Station1
Atthisstation,youwillfindtwosheetsofpaper,tape,andminimarshmallows.Eachsheetofpaper is81⁄2"by11".Youwillbeusingthesetoinvestigatevolume.
Yourtaskistoanswerthisquestion:Whichwayshouldyourollthepaperintoacylindertogetthegreatestvolume?With11inchesasthecircumferenceortheheight?Doesitmatter?
Tapeonesheetofpaperintoacylindersothat11inchesisthecircumference.Tapeonesheetofpaperintoacylindersothat11inchesistheheight.
Whichdoyouthinkholdsmore?______
Fillbothcylinderswithminimarshmallows.
Whichcylinderholdsmore?______
Nowcalculatethevolumeofthetwocylinders.Writeyoursolutionbelow.
Volumeofcylinderwith11"circumference:______Volumeofcylinderwith11"height:______
Whichcylinderhasagreatervolume?______
VolumeofCylinders,Cones,andSpheres
Station2
Atthisstation,youwillpretendyouareanicecreamvendor.Youhaveacoolerthatis3feetby2feetby2feet.Youwanttofitinasmanyicecreamconesaspossible.Eachicecreamconeispre-wrappedandhasicecreaminit,soyoucannotfittheminsideoneanother.
Thewaytofindthevolumeofaconeisthesameasfindingthevolumeofapyramid:
In the caseofthecone,thebaseisnotfoundbymultiplyingthelengthtimesthewidth.
Howdowefindtheareaofthebaseofacone?
Eachicecreamconeis4inchestall,andtheradiusis1inch.
Whatisthevolumeofoneicecreamcone?______
Whatisthevolumeofthecooler?______
Howmanyicecreamconescanyoufitinthecooler?(Becarefulofyourunits!)
Explainhowyouarrivedatyoursolution.______
VolumeofCylinders,Cones,andSpheres
Station3
Atthisstation,youwillfindacalculator,measuringtape,andseveralroundobjects.Workasagrouptodetermineeachobject’sapproximatevolumebyfollowingthestepsbelow.
1.First,usethemeasuringtapetomeasurearoundthemiddleofeachobject.Thisisthe
circumference(C).
2.Usethisvariationoftheformulaforcircumferenceofacircletofindtheradius(r)oftheobject.
Radiusofacircle
3.Finally,substitutetheradiusintotheformulaforthevolumeofasphere:
Volumeofasphere:
4.Usetheinformationtofillinthetable.
VolumeofCylinders,Cones,andSpheres
Station4
Workasagrouptosolvethefollowingproblems.Youmayuseacalculator.Roundtothenearesthundredthifnecessary.
π≈3.14
Forproblems1–4,findthevolumeofacylinderusingthegiveninformation.
Volumeofacylinder:πr2h
1.Radius=8cm;height=7cm
Volume=______
2.Radius=6in.;height=16in.
Volume=______
3.Radius=36yd;height=13yd
Volume=______
4.Radius=18mm;height=45mm
Volume=
Station 4 continued
Forproblems5–8,findthevolumeofaconeusingthegiveninformation.
1
Volumeofacone:V=πr2h
3
5.Radius=3cm;height=11cm
Volume=______
6.Radius=8mm;height=12mm
Volume=______
7.Radius=9in.;height=16in.
Volume=
8.Radius=12yd;height=20yd
Volume=______
continued
Station 4 continued
Forproblems9–12,findthevolumeofasphereusingthegiveninformation.
Volumeofasphere:
9.Radius=9ft
Volume=
10.Radius=8mm
Volume=______
11.Radius=4cm
Volume=______
12.Radius=3yd
Volume=______