CP1Math2 Name_______________________________Unit8:S.A.,Volume,Trigonometry:Day7
SurfaceArea
Objectives:• Defineimportantvocabularyfor3-dimensionalfigures• Findthesurfaceareaforvariousprisms• Generalizeaformulaforsurfaceareaofcylinders
Vocabulary
PRISM: aprismisasolidfigurewhosetwoendfacesarecongruentandparallelpolygons,andwhosesidesareparallelograms.
Whichshapesbelowareprisms??
CYLINDER:acylinderisa3-dimensionalobjectwith2circular,parallelfaces.
Rightcylinder ObliquecylinderPYRAMID:a3-dimensionalfigurewithapolygonforabaseandwithasinglevertex(the“apex”),whichisnotinthesameplaneasthebase
TriangularpyramidSquarepyramidObliquesquarepyramid PentagonalpyramidHexagonalPyramid CONE:a3-dimensionalfigurewithacicleforabaseandwithasinglevertexpointnotinthesameplaneasthebase Rightcone: Obliquecone:**3-dimensionalfigurescanbe“right”(perpendicular)or“oblique”asshownabove.SPHERE:a3-dimensionalfigureinwhicheverypointonthesurfaceisequidistancefromthecenter
SURFACEAREAisthetotalareaofthesurfaces(orfaces)ofa3-dimensionalobject.Sometimesweflattena3-dimensionalfigureintoa2-dimensionalnettofinditssurfacearea. Hereisanetofacube(prismwithallsidescongruent): Wemeasuresurfaceareainsquareunits(cm2,in2,ft2,etc)becausewearemeasuringin2dimensions.Tofindthesurfaceareaofprismsandpyramids,considerthenetofallthefaces.Findtheareaofallthefacesseparatelythenaddtheareastogether.Tofindthesurfaceareaofcylinders,cones,andspheres,weusuallyrelyontheformulasgivenbelow.Thelastpageofthispacketshowsyouwheresomeoftheformulascomefromandareleftasanoptionalchallenge.
SurfaceAreaofaCylinder SurfaceAreaofaCone SurfaceAreaofaSphere
𝑆𝐴 = 2𝜋𝑟! + 2𝜋𝑟ℎ 𝑆𝐴 = 𝜋𝑟! + 𝜋𝑟𝑙 𝑆𝐴 = 4𝜋𝑟!
SLANTHEIGHT(𝒍):thedistancemeasuredalongthelateralface(oneofthesides)fromthebasetotheapex
Practice1. Findthesurfaceareaofthefollowingshapes.Besuretoincludeunitsinyouranswerswhenappropriate.
a. Arectangularprismthatmeasures1cmby2cmby4cm
b. Atriangularprismwiththegivendimensions:
2cm
4cm
1cm
(Intheimagesontheright,histheheightandlistheslantheight)
c. Anopen-toptriangularprism,asshownbelow.Giveanexactanswer(nodecimals).
d. Theobliquecylindershownbelow.
2. Findthetotalsurfacearea(includingtherectangularface)ofahalfcylinderwitharadiusof5inchesandaheightof2inches,asshownbelow.
3. Charlie’snewhathasthedimensionsshown.Findtheamountofclothneededtomakethehat. open
30cm
32cm
4. Aconefitsperfectlyinsideacylinderwithaheightof7cm,asshownbelow.a. Findthesurfaceareaofthecylinder.b. Findthesurfaceareaofthecone.
5. Findthesurfaceareaofthesolidshownbelow.
Answers1. a. 28cm2 b. 120in2 c. 144+ 66 3 d. 1170𝜋2. 20+ 35𝜋in23. 544𝜋cm24. a. 116.7𝜋 ≈ 366.5cm2 b. 65.9𝜋 ≈ 207cm2
5. 93𝜋
55°
7 cm
OptionalChallengeProblemsFindingtheformulasforcylindersandcones1. Hereyouaregiventhenetofageneralcylinderwithbaseradiusrandheighth.Noticethatitismadeup
oftwocirclesandarectangle.
a. Whatistheareaofoneofthecircles(thebaseofthecylinder)?
b. Weknowtheheightoftherectangleish.Whatisthewidthoftherectangle?
c. Whatistheareaoftherectangle?
d. Whatisthetotalsurfaceareaofthecylinder?2. Belowisthenetofarightcircularcone.Noticethatitismadeupofacircleandasectorofacircle.
a. Whatistheareaofthecircle?
b. Whatistheradiusofthesector?
c. Whatisthearclengthofthesector,intermsofr?
d. Knowingthearclengthandtheradiusofthesector,findtheanglemeasureorθofthesector.
e. Nowthatweknowtheradiusandanglemeasureofthesector,findtheareaofthesector.
f. Sincetheconeismadeupofthesetwoshapes,whatisthesurfaceareaformulaforarightcircularcone?