gre geometry

6/1/2015 SparkNotes:GRE:Geometry

http://www.sparknotes.com/testprep/books/gre/chapter2section3.rhtml 1/17

Math101

2.1 ARITHMETIC2.2 ALGEBRA

2.3 GEOMETRY2.4 DATAANALYSIS

GeometryGeometryhasalongandstoriedhistorythatgoesbackthousandsofyears,whichweresureyouredyingtohearabout.ButunlessyouretakingtheGREtogoforyourmastersinmathhistory,thishasnopossiblerelevancetoyourlife.Sowellskipthehistorylessonandgetrighttothefacts.

LinesandAnglesAnangleiscomposedoftwolinesandisameasureofthespreadbetweenthem.Thepointwherethetwolinesmeetiscalledthevertex.OntheGRE,anglesaremeasuredindegrees.Heresanexample:

Theafterthe45isadegreesymbol.Thisanglemeasures45degrees.

Acute,Obtuse,andRightAnglesAnglesthatmeasurelessthan90arecalledacuteangles.Anglesthatmeasuremorethan90arecalledobtuseangles.AspecialanglethattheGREtestmakersloveiscalledarightangle.Rightanglesalwaysmeasureexactly90andareindicatedbyalittlesquarewheretheanglemeasurewouldnormallybe,likethis:

StraightAnglesMultipleanglesthatmeetatasinglepointonalinearecalledstraightangles.Thesumoftheanglesmeetingatasinglepointonastraightlineisalwaysequalto180.Youmayseesomethinglikethis,askingyoutosolveforn:

Thetwoangles(onemarkedby40andtheonemarkedbyn)meetatthesamepointontheline.Sincethesumoftheanglesonthelinemustbe180,youcanplug40and

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nintoaformulalikethis:

40+n=180

Subtracting40frombothsidesgivesn=140,orn=140.

VerticalAnglesWhentwolinesintersect,theanglesthatlieoppositeeachother,calledverticalangles,arealwaysequal.

Angles and areverticalanglesandarethereforeequal.Angles andarealsovertical,equalangles.

ParallelLinesAmorecomplicatedversionofthisfigurethatyoumayseeontheGREinvolvestwoparallellinesintersectedbyathirdline,calledatransversal.Parallelmeansthatthelinesruninexactlythesamedirectionandneverintersect.Heresanexample:

line1isparalleltoline2

Dontassumelinesareparalleljustbecausetheylookliketheyarethequestionwillalwaystellyouiftwolinesaremeanttobeparallel.Eventhoughthisfigurecontainsmanyangles,itturnsoutthatitonlyhastwokindsofangles:bigangles(obtuse)andlittleangles(acute).Allthebiganglesareequaltoeachother,andallthelittleanglesareequaltoeachother.Furthermore,anybigangle+anylittleangle=180.Thisistrueforanyfigurewithtwoparallellinesintersectedbyathirdline.Inthefollowingfigure,welabeleveryangletoshowyouwhatwemean:

Asdescribedabove,theeightanglescreatedbythesetwointersectionshavespecialrelationshipstoeachother:

Angles1,4,5,and8areequaltooneanother.Angle1isverticaltoangle4,andangle5isverticaltoangle8.Angles2,3,6,and7areequaltooneanother.Angle2isverticaltoangle3,andangle6isverticaltoangle7.Thesumofanytwoadjacentangles,suchas1and2or7and8,equals180becausetheyformastraightanglelyingonaline.Thesumofanybigangle+anylittleangle=180,sincethebigandlittleanglesinthisfigurecombineintostraightlinesandallthebiganglesareequalandallthelittleanglesareequal.Soabigandlittleangledontneedtobenexttoeachothertoaddto180anybigplusanylittlewilladdto180.Forexample,sinceangles1and2sumto180,andsinceangles2and7areequal,thesumofangles1and7alsoequals180.

Byusingtheserules,youcanfigureoutthedegreesofanglesthatmayseemunrelated.Forexample:

Ifline1isparalleltoline2,whatisthevalueofyinthefigureabove?



Again,thisfigurehasonlytwokindsofangles:bigangles(obtuse)andlittleangles(acute).Weknowthat55isalittleangle,soymustbeabigangle.Sincebigangle+littleangle=180,youcanwrite:

y+55=180

Solvingfory:

y=18055

y=125

PerpendicularLinesTwolinesthatmeetatarightanglearecalledperpendicularlines.Ifyouretoldtwolinesareperpendicular,justthink90.Wedlovetotellyoumoreaboutthesebeauties,buttheresreallynotmuchelsetosay.

PolygonBasicsApolygonisatwodimensionalfigurewiththreeormorestraightsides.Polygonsarenamedaccordingtothenumberofsidestheyhave.

NumberofSides Name

3 triangle

4 quadrilateral

5 pentagon

6 hexagon

7 heptagon

8 octagon

9 nonagon

10 decagon

12 dodecagon

n ngon

Allpolygons,nomatterhowmanysidestheypossess,sharecertaincharacteristics:

Thesumoftheinterioranglesofapolygonwithnsidesis(n2)180.Forinstance,thesumoftheinterioranglesofanoctagonis(82)180=6(180)=1080.Apolygonwithequalsidesandequalinterioranglesisaregularpolygon.Thesumoftheexterioranglesofanypolygonis360.Theperimeterofapolygonisthesumofthelengthsofitssides.Theareaofapolygonisthemeasureoftheareaoftheregionenclosedbythepolygon.EachpolygontestedontheGREhasitsownuniqueareaformula,whichwellcoverbelow.

Forthemostpart,thepolygonstestedinGREmathincludetrianglesandquadrilaterals.Alltriangleshavethreesides,buttherearespecialtypesoftrianglesthatwellcovernext.Thenwellmoveontofourcommonquadrilaterals(foursidedfigures)thatappearonthetest:rectangles,squares,parallelograms,andtrapezoids.

Wellthenleavetheworldofpolygonsandmakeourwaytocirclesandthenconcludethisgeometrysectionwithadiscussionofthethreedimensionalsolidsyoumayseeonyourtest:rectangularsolids,cubes,andrightcircularcylinders.

TrianglesOfallthegeometricshapes,trianglesareamongthemostcommonlytestedontheGRE.ButsincetheGREtendstotestthesametrianglesoverandover,youjustneedtomasterafewrulesandafewdiagrams.Welllookatsomespecialtrianglesshortly,butfirstwellexplainfourveryspecialrules.

TheFourRulesofTrianglesCommitthesefourrulestomemory.

1.TheRuleofInteriorAngles.Thesumoftheinterioranglesofatrianglealwaysequals180.Interioranglesarethoseontheinside.Wheneveryouregiventwoanglesofatriangle,youcanusethisformulatocalculatethethirdangle.Forexample:



Whatisthevalueofainthefigureabove?

Sincethesumoftheanglesofatriangleequals180,youcansetupanequation:

a+100+60=180

Isolatingthevariableandsolvingforagivesa=20.

2.TheRuleofExteriorAngles.Anexteriorangleofatriangleistheangleformedbyextendingoneofthesidesofthetrianglepastavertex(ortheintersectionoftwosidesofafigure).Inthefigurebelow,distheexteriorangle.

Since,together,dandcformastraightangle,theyaddupto180:d+c=180.Accordingtothefirstruleoftriangles,thethreeanglesofatrianglealwaysaddupto180,soa+b+c=180.Sinced+c=180anda+b+c=180,dmustbeequaltoa+b(theremoteinteriorangles).Thisgeneralizestoalltrianglesasthefollowingrules:Theexteriorangleofatriangleplustheinterioranglewithwhichitsharesavertexisalways180.Theexteriorangleisalsoequaltothesumofthemeasuresoftheremoteinteriorangles.

3.TheRuleoftheSides.Thelengthofanysideofatrianglemustbegreaterthanthedifferenceandlessthanthesumoftheothertwosides.Inotherwords:

differenceofothertwosides



Whatisonepossiblevalueofxif?(A) 4(B) 5(C) 7(D) 10(E) 15

Accordingtotheruleofproportion,thelongestsideofatriangleisoppositethelargestangle,andtheshortestsideofatriangleisoppositethesmallestangle.ThequestiontellsusthatangleC



rundownthestreetandproclaimittotheworld,entirelyinthebuff.OrmaybethatwasArchimedes...?Hey,itwasalongtimeago.SufficeittosaythatsomenakedGreekguythoughtupsomethingcool.

SincePythagorastookthetroubletoinventhistheorem,theleastwecandoislearnit.Besides,itsoneofthemostfamoustheoremsinallofmath,anditistestedwithregularityontheGRE,toboot.Hereitis:

Inarighttriangle,a2+b2=c2,wherecisthelengthofthehypotenuseandaandbarethelengthsofthetwolegs.

Andheresasimpleapplication:

Whatisthevalueofbinthetriangleabove?

Thelittlesquareinthelowerleftcornerletsyouknowthatthisisarighttriangle,soyourecleartousethePythagoreantheorem.Substitutingtheknownlengthsintotheformulagives:

32+b2=52

9+b2=25

b2=16

b=4

Thisisthereforetheworldfamous345righttriangle.ThankstothePythagoreantheorem,ifyouknowthemeasuresofanytwosidesofarighttriangle,youcanalwaysfindthethird.Eureka!indeed.

PythagoreanTriples.BecauserighttrianglesobeythePythagoreantheorem,onlyaspecificfewhavesidelengthsthatareallintegers.Forexample,arighttrianglewithlegsoflength3and5hasahypotenuseoflength .PositiveintegersthatobeythePythagoreantheoremarecalledPythagoreantriples,andthesearetheonesyourelikelytoseeonyourtestasthelengthsofthesidesofrighttriangles.Herearesomecommonones:

{3,4,5}

{5,12,13}

{7,24,25}

{8,15,17}

InadditiontothesePythagoreantriples,youshouldalsowatchoutfortheirmultiples.Forexample,{6,8,10}isaPythagoreantriple,sinceitisamultipleof{3,4,5},derivedfromsimplydoublingeachvalue.Thisknowledgecansignificantlyshortenyourworkinaproblemlikethis:

Whatisthevalueofzinthetriangleabove?

Sure,youcouldcalculateitoutusingthePythagoreantheorem,butwhowantstosquare120and130andthenworktheresultsintotheformula?Pythagorashimselfwouldprobablysay,Ah,screwit...andheadoffforachatwithSocrates.(Actually,PythagorasdiedabouttwentyyearsbeforeSocrateswasborn,butyougetthepoint.)

ButarmedwithourPythagoreantriples,itsnoproblemforus.Thehypotenuseis130,andoneofthelegsis120.Theratiobetweenthesesidesis130:120,or13:12.Thisexactlymatchesthe{5,12,13}Pythagoreantriple.Soweremissingthe5partofthetriplefortheotherleg.However,sincethesidesofthetriangleinthequestionare10



timeslongerthanthoseinthe{5,12,13}triple,themissingsidemustbe510=50.

306090RIGHTTRIANGLESAsyoucansee,righttrianglesareprettydarnspecial.ButtherearetwoextraspecialonesthatappearwithastoundingfrequencyontheGRE.Theyare306090righttrianglesand454590righttriangles.Whenyouseeoneofthese,insteadofworkingoutthePythagoreantheorem,youllbeabletoapplystandardratiosthatexistbetweenthelengthofthesidesofthesetriangles.

Theguywhonamed306090trianglesdidnthavemuchofanimaginationormaybehejustdidnthaveacoolnamelikePythagoras.Thenamederivesfromthefactthatthesetriangleshaveanglesof30,60,and90.So,whatssospecialaboutthat?This:Thesidelengthsof306090trianglesalwaysfollowaspecificpattern.Iftheshortlegoppositethe30anglehaslengthx,thenthehypotenusehaslength2x,andthelongleg,oppositethe60angle,haslength .Therefore:

Thesidesofevery306090trianglewillfollowtheratio1: :2

Thankstothisconstantratio,ifyouknowthelengthofjustonesideofthetriangle,youllimmediatelybeabletocalculatethelengthsoftheothertwo.If,forexample,youknowthatthesideoppositethe30angleis2meters,thenbyusingtheratioyoucandeterminethatthehypotenuseis4meters,andthelegoppositethe60angleis

meters.

454590RIGHTTRIANGLESA454590righttriangleisatrianglewithtwoanglesof45andonerightangle.Itssometimescalledanisoscelesrighttriangle,sinceitsbothisoscelesandright.Likethe306090triangle,thelengthsofthesidesofa454590trianglealsofollowaspecificpattern.Ifthelegsareoflengthx(thelegswillalwaysbeequal),thenthehypotenusehaslength .

Thesidesofevery454590trianglewillfollowtherationof

Thisratiowillhelpyouwhenfacedwithtriangleslikethis:

Thisrighttrianglehastwoequalsides,whichmeansthetwoanglesotherthantherightanglemustbe45each.Sowehavea454590righttriangle,whichmeanswecanemploythe ratio.Butinsteadofbeing1and1,thelengthsofthelegsare5and5.Sincethelengthsofthesidesinthetriangleabovearefivetimesthelengthsin

the righttriangle,thehypotenusemustbe ,or .

AreaofaTriangleTheformulafortheareaofatriangleis:

area= or

Keepinmindthatthebaseandheightofatrianglearenotjustanytwosidesofatriangle.Thebaseandheightmustbeperpendicular,whichmeanstheymustmeetatarightangle.

Letstryanexample.Whatstheareaofthistriangle?

Notethattheareaisnot ,becausethosetwosidesdonotmeetatarightangle.Tocalculatethearea,youmustfirstdetermineb.Youllprobablynoticethatthisisthe345righttriangleyousawearlier,sob=4.Nowyouhavetwoperpendicularsides,soyoucancorrectlycalculatetheareaasfollows:

Trianglesaresurelyimportant,buttheyarenttheonlygeometricfiguresyoullcomeacrossontheGRE.Wemovenowtoquadrilaterals,whicharefoursidedfigures.The



firsttwowecoveryourenodoubtfamiliarwith,nomatterhowlongyouvebeenoutofhighschool.Theothertwoyoumayhaveforgottenaboutlongago.

RectanglesArectangleisaquadrilateralinwhichtheoppositesidesareparallelandtheinterioranglesareallrightangles.Theoppositesidesofarectangleareequal,asindicatedinthefigurebelow:

AreaofaRectangleTheformulafortheareaofarectangleis:

area=baseheightorsimplya=bh

Sincethebaseisthelengthoftherectangleandtheheightisthewidth,justmultiplythelengthbythewidthtogettheareaofarectangle.

DiagonalsofaRectangleThetwodiagonalsofarectanglearealwaysequaltoeachother,andeitherdiagonalthroughtherectanglecutstherectangleintotwoequalrighttriangles.Inthefigurebelow,thediagonalBDcutsrectangleABCDintocongruentrighttrianglesBADandBCD.Congruentmeansthatthosetrianglesareexactlyidentical.

Sincethediagonaloftherectangleformsrighttrianglesthatincludethediagonalandtwosidesoftherectangle,ifyouknowtwoofthesevalues,youcanalwayscalculatethethirdwiththePythagoreantheorem.Forexample,ifyouknowthesidelengthsoftherectangle,youcancalculatethelengthofthediagonal.Ifyouknowthediagonalandonesidelength,youcancalculatetheothersidelength.Also,keepinmindthatthediagonalmightcuttherectangleintoa306090triangle,inwhichcaseyoucouldusethe1: :2ratiotomakeyourcalculatingjobeveneasier.

SquaresHey,buddy,dontyoubenosquare...

ElvisPresley,JailhouseRock

Wedontknowwhy,butsometimearoundthe1950sthewordsquarebecamesynonymouswithuncool.Ofcourse,thatsthesamedecadethatbroughtuswordslikedaddyOanddancescalledTheHandJive.

Wethinkthatsquaresareinfactniftylittlegeometriccreatures.They,alongwithcircles,areperhapsthemostsymmetricalshapesintheuniversenothingtosneezeat.Asquareissosymmetricalbecauseitsanglesareall90andallfourofitssidesareequalinlength.Rectanglesarefairlycommon,butaperfectsquareissomethingtobehold.Likearectangle,asquaresoppositesidesareparallelanditcontainsfourrightangles.Butsquaresoneuprectanglesbyvirtueoftheirequalsides.

Asifyouveneverseenoneoftheseinyourlife,hereswhatasquarelookslike:

AreaofaSquareTheformulafortheareaofasquareis:

area=s2

Inthisformula,sisthelengthofaside.Sincethesidesofasquareareallequal,all



youneedisonesidetofigureoutasquaresarea.

DiagonalsofaSquareThesquarehastwomorespecialqualities:

Diagonalsbisecteachotheratrightanglesandareequalinlength.Diagonalsbisectthevertexanglestocreate45angles.(Thismeansthatonediagonalwillcutthesquareintotwo454590triangles,whiletwodiagonalsbreakthesquareintofour454590triangles.)

Becauseadiagonaldrawnintothesquareformstwocongruent454590triangles,ifyouknowthelengthofonesideofthesquare,youcanalwayscalculatethelengthofthediagonal:

Sincedisthehypotenuseofthe454590trianglethathaslegsoflength5,accordingtotheratio ,youknowthat .Similarly,ifyouknowonlythelengthofthediagonal,youcanusethesameratiotoworkbackwardtocalculatethelengthofthesides.

ParallelogramsAparallelogramisaquadrilateralwhoseoppositesidesareparallel.Thatmeansthatrectanglesandsquaresqualifyasparallelograms,butsodofoursidedfiguresthatdontcontainrightangles.

Inaparallelogram,oppositesidesareequalinlength.Thatmeansthatinthefigureabove,BC=ADandAB=DC.Oppositeanglesareequal:ABC=ADCandBAD=BCD.Adjacentanglesaresupplementary,whichmeanstheyaddupto180.Here,anexampleisABC+BCD=180.

AreaofaParallelogramTheareaofaparallelogramisgivenbytheformula:

area=bh

Inthisformula,bisthelengthofthebase,andhistheheight.Asshowninthefigurebelow,theheightofaparallelogramisrepresentedbyaperpendicularlinedroppedfromonesideofthefiguretothesidedesignatedasthebase.

DiagonalsofaParallelogram

Thediagonalsofaparallelogrambisect(split)eachother:BE=EDandAE=ECOnediagonalsplitsaparallelogramintotwocongruenttriangles:ABD=BCDTwodiagonalssplitaparallelogramintotwopairsofcongruenttriangles:AEB=DECandBEC=AED

TrapezoidsAtrapezoidmaysoundlikeanewStarWarscharacter,butitsactuallythenameofaquadrilateralwithonepairofparallelsidesandonepairofnonparallelsides.Heresanexample:



Inthistrapezoid,ABisparalleltoCD(shownbythearrowmarks),whereasACandBDarenotparallel.

AreaofaTrapezoidTheformulafortheareaofatrapezoidisabitmorecomplexthantheareaformulasyouveseenfortheotherquadrilateralsinthissection.Wellsetitoffsoyoucantakeagoodlookatit:

Inthisformula,s1ands2arethelengthsoftheparallelsides(alsocalledthebasesofthetrapezoid),andhistheheight.Inatrapezoid,theheightistheperpendiculardistancefromonebasetotheother.

IfyoucomeacrossatrapezoidquestionontheGRE,youmayneedtouseyourknowledgeoftrianglestosolveit.Heresanexampleofwhatwemean:

Findtheareaofthefigureabove.

Firstofall,werenottoldthatthefigureisatrapezoid,butwecaninferasmuchfromtheinformationgiven.Sinceboththelinelabeled6andthelinelabeled10formrightangleswiththelineconnectingthem,the6and10linesmustbeparallel.Meanwhile,theothertwolines(theleftandrightsidesofthefigure)cannotbeparallelbecauseoneconnectstothebottomlineatarightangle,whiletheotherconnectswiththatlineata45angle.Sowecandeducethatthefigureisatrapezoid,whichmeansthetrapezoidareaformulaisinplay.

Thebasesofthetrapezoidaretheparallelsides,andweretoldtheirlengthsare6and10.Sofarsogood,buttofindthearea,wealsoneedtofindthetrapezoidsheight,whichisntgiven.

Todothat,splitthetrapezoidintoarectangleanda454590trianglebydrawingintheheight.

Onceyouvedrawnintheheight,youcansplitthebasethatsequalto10intotwoparts:Thebaseoftherectangleis6,andthelegofthetriangleis4.Sincethetriangleis454590,thetwolegsmustbeequal.Thisleg,though,isalsotheheightofthetrapezoid.Sotheheightofthetrapezoidis4.Nowyoucanplugthenumbersintotheformula:

Anotherwaytofindtheareaofthetrapezoidistofindtheareasofthetriangleandtherectangle,thenaddthemtogether:

area= (44)+(64)

(16)+24

8+24=32

CirclesAcircleisthecollectionofpointsequidistantfromagivenpoint,calledthecenter.Circlesarenamedaftertheircenterpointsthatsjusteasierthangivingthemnames



likeRalphandBetty.Allcirclescontain360.Thedistancefromthecentertoanypointonthecircleiscalledtheradius(r).Radiusisthemostimportantmeasurementinacircle,becauseifyouknowacirclesradius,youcanfigureoutallitsothercharacteristics,suchasitsarea,diameter,andcircumference.Wellcoverallthatinthenextfewpages.Thediameter(d)ofacirclestretchesbetweenendpointsonthecircle,passingthroughthecenter.Achordalsoextendsfromendpointtoendpointonthecircle,butitdoesnotnecessarilypassthroughthecenter.Inthefollowingfigure,pointCisthecenterofthecircle,ristheradius,andABisachord.

TangentLinesTangentsarelinesthatintersectacircleatonlyonepoint.Justlikeeverythingelseingeometry,tangentlinesaredefinedbycertainfixedrules.

Heresthefirst:Aradiuswhoseendpointistheintersectionpointofthetangentlineandthecircleisalwaysperpendiculartothetangentline,asshowninthefollowingfigure:

Andthesecondrule:Everypointinspaceoutsidethecirclecanextendexactlytwotangentlinestothecircle.Thedistancesfromtheoriginofthetwotangentstothepointsoftangencyarealwaysequal.Inthefigurebelow,XY=XZ.

CentralAnglesandInscribedAnglesAnanglewhosevertexisthecenterofthecircleiscalledacentralangle.

Thedegreeofthecircle(thesliceofpie)cutbyacentralangleisequaltothemeasureoftheangle.Ifacentralangleis25,thenitcutsa25arcinthecircle.

Aninscribedangleisanangleformedbytwochordsoriginatingfromasinglepoint.

Aninscribedanglewillalwayscutoutanarcinthecirclethatistwicethesizeofthedegreeoftheinscribedangle.Forexample,ifaninscribedangleis40,itwillcutanarcof80inthecircle.



Ifaninscribedangleandacentralanglecutoutthesamearcinacircle,thecentralanglewillbetwiceaslargeastheinscribedangle.

CircumferenceofaCircleThecircumferenceistheperimeterofthecirclethatis,thetotaldistancearoundthecircle.Theformulaforcircumferenceofacircleis:

circumference=2r

Inthisformula,ristheradius.Sinceacirclesdiameterisalwaystwiceitsradius,theformulacanalsobewrittenc=d,wheredisthediameter.Letsfindthecircumferenceofthecirclebelow:

Pluggingtheradiusintotheformula,c=2r=2(3)=6.

ArcLengthAnarcisapartofacirclescircumference.Anarccontainstwoendpointsandallthepointsonthecirclebetweentheendpoints.Bypickinganytwopointsonacircle,twoarcsarecreated:amajorarc,whichisbydefinitionthelongerarc,andaminorarc,theshorterone.

Sincethedegreeofanarcisdefinedbythecentralorinscribedanglethatinterceptsthearcsendpoints,youcancalculatethearclengthaslongasyouknowthecirclesradiusandthemeasureofeitherthecentralorinscribedangle.

Thearclengthformulais:

arclength=

Inthisformula,nisthemeasureofthedegreeofthearc,andristheradius.Thismakessense,ifyouthinkaboutit:Thereare360inacircle,sothedegreeofanarcdividedby360givesusthefractionofthetotalcircumferencethatarcrepresents.Multiplyingthatbythetotalcircumference(2pr)givesusthelengthofthearc.

Heresthesortofarclengthquestionyoumightseeonthetest:

CircleDhasradius9.WhatisthelengthofarcAB?

TofigureoutthelengthofarcAB,weneedtoknowtheradiusofthecircleandthemeasureofC,theinscribedanglethatinterceptstheendpointsofarcAB.Thequestionprovidestheradiusofthecircle,9,butitthrowsusalittlecurveballbynotprovidingthemeasureofC.Instead,thequestionputsCinatriangleandtellsusthemeasuresoftheothertwoanglesinthetriangle.Likewesaid,onlyalittlecurveball:YoucaneasilyfigureoutthemeasureofCbecause,asyouknowbynow,thethreeanglesofatriangleaddupto180:



c=180(50+70)

c=180120

c=60

Sincecisaninscribedangle,arcABmustbetwiceitsmeasure,or120.Nowwecanplugthesevaluesintotheformulaforarclength:

AreaofaCircleIfyouknowtheradiusofacircle,youcanfigureoutitsarea.Theformulaforareais:

area=r2

Inthisformula,ristheradius.Sowhenyouneedtofindtheareaofacircle,yourrealgoalistofigureouttheradius.Ineasierquestionstheradiuswillbegiven.Inharderquestions,theyllgiveyouthediameterorcircumferenceandyoullhavetousetheformulasforthosetocalculatetheradius,whichyoullthenplugintotheareaformula.

AreaofaSectorAsectorofacircleistheareaenclosedbyacentralangleandthecircleitself.Itsshapedlikeasliceofpizza.Theshadedregioninthefigureontheleftbelowisasector.Thefigureontherightisasliceofpepperonipizza.Seetheresemblance?

Theareaofasectorisrelatedtotheareaofacirclejustasthelengthofanarcisrelatedtothecircumference.Tofindtheareaofasector,findwhatfractionof360thesectormakesupandmultiplythisfractionbythetotalareaofthecircle.Informulaform:

areaofasector=

Inthisformula,nisthemeasureofthecentralanglethatformstheboundaryofthesector,andristheradius.Anexamplewillhelp.Findtheareaofthesectorinthefigurebelow:

Thesectorisboundedbya70centralangleinacirclewhoseradiusis6.Usingtheformula,theareaofthesectoris:

MishMashes:FigureswithMultipleShapesTheGREtestmakersevidentlyfeelitwouldbenofunifallofthesegeometricshapesyourelearningappearedinisolation,sosometimestheymashthemtogether.Thetrickintheseproblemsistounderstandandbeabletomanipulatetherulesofeachfigureindividually,whilealsorecognizingwhichelementsofthemishmashesoverlap.Forexample,thediameterofacirclemayalsobethesideofasquare,soifyouusetherulesofcirclestocalculatethatlength,youcanthenusethatanswertodeterminesomethingaboutthesquare,suchasitsareaorperimeter.

Mishmashproblemsoftencombinecircleswithotherfigures.Heresanexample:



WhatisthelengthofminorarcBEincircleAiftheareaofrectangleABCDis18?

TofindthelengthofminorarcBE,youhavetoknowtwothings:theradiusofthecircleandthemeasureofthecentralanglethatintersectsthecircleatpointsBandE.BecauseABCDisarectangle,andrectanglesonlyhaverightangles,BADis90.Inthisquestion,theytellyouasmuchbyincludingtherightanglesign.Butinaharderquestion,theydleavetherightanglesignoutandexpectyoutodeducethatBADis90onyourown.AndsincethatanglealsohappenstobethecentralangleofcircleAinterceptingthearcinquestion,wecandeterminethatarcBEmeasures90.

Findingtheradiusrequiresabitofcreativevisualizationaswell,butitsnotsohard.Thekeyistorealizethattheradiusofthecircleisequaltothewidthoftherectangle.Soletsworkbackwardfromtherectangletogiveuswhatweneedtoknowaboutthecircle.Theareaoftherectangleis18,anditslengthis6.Sincetheareaofarectangleissimplyitslengthmultipliedbyitswidth,wecandivide18by6togetawidthof3.Asweveseen,thisrectanglewidthdoublesasthecirclesradius,sowereinbusiness:radius=3.Allwehavetodoispluginthevalueswefoundintothearclengthformula,andweredone.

Thatcoversthetwodimensionalfiguresyoushouldknow,buttherearealsosomethreedimensionalfiguresyoumaybeaskedaboutaswell.Wellfinishupthisgeometrysectionwithalookatthose.

RectangularSolidsArectangularsolidisaprismwitharectangularbaseandedgesthatareperpendiculartoitsbase.InEnglish,itlooksalotlikeacardboardbox.

Arectangularsolidhasthreeimportantdimensions:length(l),width(w),andheight(h).Ifyouknowthesethreemeasurements,youcanfindthesolidsvolume,surfacearea,anddiagonallength.

VolumeofaRectangularSolidTheformulaforthevolumeofarectangularsolidbuildsontheformulafortheareaofarectangle.Asdiscussedearlier,theareaofarectangleisequaltoitslengthtimesitswidth.Theformulaforthevolumeofarectangularsolidaddsthethirddimension,height,toget:

volume=lwh

Heresagoodoldfashionedexample:

Whatisthevolumeofthefigurepresentedbelow?

Thelengthis3x,thewidthisx,andtheheightis2x.Justplugthevaluesintothevolumeformulaandyouregoodtogo:v=(3x)(x)(2x)=6x3.



SurfaceAreaofaRectangularSolidThesurfaceareaofasolidistheareaofitsoutermostskin.Inthecaseofrectangularsolids,imagineacardboardboxallclosedup.Thesurfaceofthatclosedboxismadeofsixrectangles:Thesumoftheareasofthesixrectanglesisthesurfaceareaofthebox.Tomakethingseveneasier,thesixrectanglescomeinthreecongruentpairs.Wevemarkedthecongruentpairsbyshadesofgrayintheimagebelow:Onepairisclear,onepairislightgray,andonepairisdarkgray.

Twofaceshaveareasoflw,twofaceshaveareasoflh,andtwofaceshaveareasofwh.Thesurfaceareaoftheentiresolidisthesumoftheareasofthecongruentpairs:surfacearea=2lw+2lh+2wh.

Letstrytheformulaoutonthesamesolidwesawabove.Findthesurfaceareaofthis:

Again,thelengthis3x,thewidthisx,andtheheightis2x.Pluggingintotheformula,weget:

surfacearea=2(3x)(x)+2(3x)(2x)+2(x)(2x)

=6x2+12x2+4x2

=22x2

DIVIDINGRECTANGULARSOLIDSIfyouredoingreallywellontheMathsection,theCATprogramwillbeginscroungingforthetoughest,mostesotericproblemsitcanfindtothrowyourway.Onesuchproblemmaydescribeasolid,giveyouallofitsmeasurements,andthentellyouthattheboxhasbeencutinhalf,likeso:

Anumberofpossiblequestionscouldbecreatedfromthisscenario.Forexample,youmaybeaskedtofindthecombinedsurfaceareaofthetwonewboxes.OrmaybeaQuantitativeComparisonquestionwouldaskyoutocomparethevolumeoftheoriginalsolidwiththatofthetwonewones.Actually,thevolumeremainsunchanged,butthesurfaceareaincreasesbecausetwonewsides(shadedinthediagram)emergewhentheboxiscutinhalf.Youmayneedtoemployabitofreasoningalongwiththeformulasyourelearningtoansweradifficultquestionlikethis,butithelpstoknowthisgeneralrule:Wheneverasolidiscutintosmallerpieces,itssurfaceareaincreases,butitsvolumeisunchanged.

DiagonalLengthofaRectangularSolidThediagonalofarectangularsolid,d,isthelinesegmentwhoseendpointsareoppositecornersofthesolid.Everyrectangularsolidhasfourdiagonals,eachwiththesamelength,thatconnecteachpairofoppositevertices.Heresonediagonaldrawnin:

Itspossiblethataquestionwilltesttoseeifyoucanfindthelengthofadiagonal.Herestheformula:

Again,listhelength,wisthewidth,andhistheheight.TheformulaislikeapumpedupversionofthePythagoreantheorem.Checkitoutinaction:



WhatisthelengthofdiagonalAHintherectangularsolidbelowifAC=5,GH=6,andCG=3?

Thequestiongivesthelength,width,andheightoftherectangularsolid,soyoucanjustplugthosenumbersintotheformula:

Theproblemcouldbemademoredifficultifitforcedyoutofirstcalculatesomeofthedimensionsbeforepluggingthemintotheformula.

CubesAcubeisathreedimensionalsquare.Thelength,width,andheightofacubeareequal,andeachofitssixfacesisasquare.Hereswhatitlookslikeprettybasic:

VolumeofaCubeTheformulaforfindingthevolumeofacubeisessentiallythesameastheformulaforthevolumeofarectangularsolid:Wejustneedtomultiplythelength,width,andheight.However,sinceacubeslength,width,andheightareallequal,theformulaforthevolumeofacubeiseveneasier:

volume=s3

Inthisformula,sisthelengthofoneedgeofthecube.

SurfaceAreaofaCubeSinceacubeisjustarectangularsolidwhosesidesareallequal,theformulaforfindingthesurfaceareaofacubeisthesameastheformulaforfindingthesurfaceareaofarectangularsolid,exceptwithssubstitutedinforl,w,andh.Thisboilsdownto:

volume=6s2

DiagonalLengthofaCubeTheformulaforthediagonalofacubeisalsoadaptedfromtheformulaforthediagonallengthofarectangularsolid,withssubstitutedforl,w,andh.Thisyields

,whichsimplifiesto:

volume=

RightCircularCylindersArightcircularcylinderlookslikeoneofthosecardboardthingsthattoiletpapercomeson,exceptitisnthollow.Ithastwocongruentcircularbasesandlookslikethis:

Theheight,h,isthelengthofthelinesegmentwhoseendpointsarethecentersofthecircularbases.Theradius,r,istheradiusofitsbase.FortheGRE,allyouneedtoknowaboutarightcircularcylinderishowtocalculateitsvolume.

VolumeofaRightCircularCylinderThevolumeofthiskindofsolidistheproductoftheareaofitsbaseanditsheight.Becausearightcircularcylinderhasacircularbase,itsvolumeisequaltotheareaofthecircularbasetimestheheightor:

volume=r2h

gre geometry

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angles andarealsovertical