2 Surface Area and Volume

111 Pages·2014·16.66 MB·English

MAHS-DV Geometry Q4
Flexbook
Adrienne Wooten
Lori Jordan, (LoriJ)
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Printed: March11,2014
iii
Contents www.ck12.org
Contents
1 PerimeterandArea 1
1.1 TrianglesandParallelograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Trapezoids,Rhombi,andKites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 AreasofSimilarPolygons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 AreaandPerimeterofRegularPolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5 PopulationDensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 SurfaceAreaandVolume 38
2.1 ExploringSolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 SurfaceAreaofPrismsandCylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.3 SurfaceAreaofPyramidsandCones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.4 VolumeofPrismsandCylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5 VolumeofPyramidsandCones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.6 SurfaceAreaandVolumeofSpheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.7 ExploringSimilarSolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
2.8 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
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www.ck12.org Chapter1. PerimeterandArea
C 1
HAPTER
Perimeter and Area
Chapter Outline
1.1 TRIANGLES AND PARALLELOGRAMS
1.2 TRAPEZOIDS, RHOMBI, AND KITES
1.3 AREAS OF SIMILAR POLYGONS
1.4 AREA AND PERIMETER OF REGULAR POLYGONS
1.5 POPULATION DENSITY
Now that we have explored triangles, quadrilaterals, polygons, and circles, we are going to learn how to find the
perimeterandareaofeach. Firstwewillderiveeachformulaandthenapplythemtodifferenttypesofpolygonsand
circles. Inaddition,wewillexplorethepropertiesofsimilarpolygons,theirperimetersandtheirareas.
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1.1. TrianglesandParallelograms www.ck12.org
1.1 Triangles and Parallelograms
Learning Objectives
• Understandthebasicconceptsofarea.
• Useformulastofindtheareaoftrianglesandparallelograms.
Review Queue
1. Defineperimeterandarea,inyourownwords.
2. Solvetheequationsbelow. Simplifyanyradicals.
a. x2=121
b. 4x2=80
c. x2−6x+8=0
3. Ifarectanglehassides4and7,whatistheperimeter?
KnowWhat? Ed’sparentsaregettinghimanewbed. Hehasdecidedthathewouldlikeakingbed. Uponfurther
research, Ed discovered there are two types of king beds, an Eastern (or standard) King and a California King.
The Eastern King has 76(cid:48)(cid:48)×80(cid:48)(cid:48) dimensions, while the California King is 72(cid:48)(cid:48)×84(cid:48)(cid:48) (both dimensions are width ×
length). Which bed has a larger area to lie on? Which one has a larger perimeter? If Ed is 6’4”, which bed makes
moresenseforhimtobuy?
Areas and Perimeters of Squares and Rectangles
Perimeter: Thedistancearoundashape. Or,thesumofalltheedgesofatwo-dimensionalfigure.
The perimeter of any figure must have a unit of measurement attached to it. If no specific units are given (feet,
inches,centimeters,etc),write“units.”
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www.ck12.org Chapter1. PerimeterandArea
Example1: Findtheperimeterofthefiguretotheleft.
Solution: First, notice there are no units, but the figure is on a grid. Here, we can use the grid as our units.
Countaroundthefiguretofindtheperimeter. Wewillstartatthebottomleft-handcornerandgoaroundthefigure
clockwise.
5+1+1+1+5+1+3+1+1+1+1+2+4+7
Theansweris34units.
Youareprobablyfamiliarwiththeareaofsquaresandrectanglesfromapreviousmathclass. Recallthatyoumust
alwaysestablishaunitofmeasureforarea. Areaisalwaysmeasuredinsquareunits,squarefeet(ft.2),squareinches
(in.2). squarecentimeters(cm.2),etc. Makesurethatthelengthandwidthareinthesameunitsbeforeapplyingany
areaformula. Ifnospecificunitsaregiven,write“units2.”
Example2: FindtheareaofthefigurefromExample1.
Solution: Ifthefigureisnotastandardshape,youcancountthenumberofsquareswithinthefigure. Ifwestarton
theleftandcounteachcolumn,wewouldhave:
5+6+1+4+3+4+4=27units2
AreaofaRectangle: Theareaofarectangleistheproductofitsbase(width)andheight(length)A=bh.
Example3: Findtheareaandperimeterofarectanglewithsides4cmby9cm.
Solution: Theperimeteris4+9+4+9=36cm. TheareaisA=9·4=36cm2.
Inthisexampleweseethataformulacanbegeneratedfortheperimeterofarectangle.
PerimeterofaRectangle: P=2b+2h,wherebisthebase(orwidth)andhistheheight(orlength).
Ifarectangleisasquare,withsidesoflengths,theformulasareasfollows:
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1.1. TrianglesandParallelograms www.ck12.org
PerimeterofaSquare: P =2s+2s=4s
square
AreaofaSquare: A =s·s=s2
sqaure
Example4: Theareaofasquareis75in2. Findtheperimeter.
Solution: Tofindtheperimeter,weneedtofindthelengthofthesides.
A=s2=75in2
√ √
s= 75=5 3in
(cid:16) √ (cid:17) √
Fromthis, P=4 5 3 =20 3in.
Area Postulates
CongruentAreasPostulate: Iftwofiguresarecongruent,theyhavethesamearea.
Thispostulateneedsnoproofbecausecongruentfigureshavethesameamountofspaceinsidethem. However,two
figureswiththesameareaarenotnecessarilycongruent.
Example5: Drawtwodifferentrectangleswithanareaof36cm2.
Solution: Thinkofallthedifferentfactorsof36. Thesecanallbedimensionsofthedifferentrectangles.
Otherpossibilitiescouldbe6×6,2×18,and1×36.
AreaAdditionPostulate: Ifafigureiscomposedoftwoormorepartsthatdonotoverlapeachother,thenthearea
ofthefigureisthesumoftheareasoftheparts.
Example6: Findtheareaofthefigurebelow. Youmayassumeallsidesareperpendicular.
Solution: Splittheshapeintotworectanglesandfindtheareaofeach.
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www.ck12.org Chapter1. PerimeterandArea
A =6·2=12 ft2
toprectangle
A =3·3=9 ft2
bottomsquare
Thetotalareais12+9=21 ft2.
Area of a Parallelogram
Recallthataparallelogramisaquadrilateralwhoseoppositesidesareparallel.
Tofindtheareaofaparallelogram,makeitintoarectangle.
Fromthis,weseethattheareaofaparallelogramisthesameastheareaofarectangle.
AreaofaParallelogram: TheareaofaparallelogramisA=bh.
Becareful! Theheightofaparallelogramisalwaysperpendiculartothebase. Thismeansthatthesidesarenotthe
height.
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1.1. TrianglesandParallelograms www.ck12.org
Example7: Findtheareaoftheparallelogram.
Solution: A=15·8=120in2
Example8: Iftheareaofaparallelogramis56units2 andthebaseis4units,whatistheheight?
Solution: Pluginwhatweknowtotheareaformulaandsolvefortheheight.
56=4h
14=h
Area of a Triangle
Ifwetakeparallelogramandcutitinhalf,alongadiagonal,wewouldhavetwocongruenttriangles. Therefore,the
formulafortheareaofatriangleisthesameastheformulaforareaofaparallelogram,butcutinhalf.
AreaofaTriangle: A= 1bhorA= bh.
2 2
Inthecasethatthetriangleisarighttriangle,thentheheightandbasewouldbethelegsoftherighttriangle. Ifthe
triangleisanobtusetriangle,thealtitude,orheight,couldbeoutsideofthetriangle.
Example9: Findtheareaandperimeterofthetriangle.
Solution: Thisisanobtusetriangle. First, tofindthearea, weneedtofindtheheightofthetriangle. Wearegiven
the two sides of the small right triangle, where the hypotenuse is also the short side of the obtuse triangle. From
thesevalues,weseethattheheightis4becausethisisa3-4-5righttriangle. TheareaisA= 1(4)(7)=14units2.
2
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