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?