2n )" Tehani"drew"this"diagram"as"part"of"her"work"as"she"developed"this" formula."""
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1 26 7.5FromPolygonstoCircles ASolidifyUnderstandingTask Part&1:&From&perimeter&to&circumference& Intheprevioustask,PlanningtheGazebo,youdevelopeda strategyforfindingtheperimeterofaregularpolygonwithn sidesinscribedinacircleofradiusr.tehani sstrategy consistsofthefollowingformula: 2013www.flickr.com/photos/andy4li P= rsin 360 ) Tehanidrewthisdiagramaspartofherworkasshedevelopedthis formula. 1. UsingTehani sdiagram,explainindetailhowshearrivedather formula. 2. Sincenistheonlythingthatvariesinthisformula,TravissuggeststhatTehanimight rewriteherformulaintheform P = 2r[ n sin 360 )].BecausetheperimeterofannSgon approximatesthecircumferenceofacirclewhennisalargenumberofsides,travis suggeststheyexaminewhathappenstothe n sin 360 )portionoftehani sformulaasn getslargerandlarger.useacalculatororspreadsheettocompletethefollowingtableto seewhathappens. n n sin 360 ) ,000 10, WriteaformulaforthecircumferenceofacirclebasedonTehani sformulaforthe perimeterofaninscribedregularnsgonandwhatyouhaveobservedwhilegeneratingthis table. 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
2 27 Part&2:&From&the&area&of&a&polygon&to&the&area&of&a&circle& Approach#1 Tehani sformulafortheareaofaregularpolygonwithnsidesinscribedinacircleofradiusris: A = n rsin 360 ) rcos 360 ) 4. ExplainindetailhowTehaniarrivedatthisformula.Youmayrefertothediagramabove. 5. TravissuggeststhattheymightrewriteTehani sformulaintheform A = r 2 [ n sin 360 ) cos 360 )]andthenexaminewhathappenstothelastpartofthe formulaasngetslargerandlarger.useacalculatororspreadsheettocompletethe followingtableandseewhathappens. n n sin ,000 10,000 ) cos 360 ) 6. WriteaformulafortheareaofacirclebasedonTehani sformulafortheareaofan inscribedregularnsgonandwhatyouhaveobservedwhilegeneratingthistable. 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
3 28 Approach#2 Acirclecanbedecomposedintoasetofthin,concentricrings,asshownontheleftinthefollowing diagram.ifweunrollandstacktheseringswecanapproximateatriangleasshowninthefigureon theright. 7. Howmightwedescribetheheightofthis triangle relativetothecircle? 8. Howmightwedescribethelengthofthebaseofthis triangle relativetothecircle? 9. Astheringsgetnarrowerandnarrowerthetriangularshapegetscloserandclosertoan exacttrianglewiththesameareaasthecircle.whatwouldthisdiagramsuggestforthe formulaoftheareaofacircle? Approach#3 Acirclecanbedecomposedintoasetofcongruentsectors,asshownontheleftinthefollowing diagram.wecanrearrangethesesectorstoapproximateaparallelogramasshowninthefigureon theright. 10. Howmightwedescribetheheightofthe parallelogram relativetothecircle? 11. Howmightwedescribethebaseofthis parallelogram relativetothecircle? 12. Aswedecomposethecircleintomoreandmoresectorsthe parallelogram shapegets closerandclosertoanexactparallelogramwiththesameareaasthecircle.whatwould thisdiagramsuggestfortheformulafortheareaofacircle? 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.
4 Circles:)a)Geometric)Perspective)) 7.5) 2013MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense Ready,Set,Go Ready Topic:AnglesandArcsofcircles,ratioswithsimilarshapes Findtheindicatedvaluesgiventhediagramandmeasurementsprovidedbelow. 1.Giventhat LGHand GHJareboth45 Whatothermeasurementofanglesorarcsdoyouknow? Listthembelowtrytofindsix) 2.Giventhat GKHhastwosidesthatareradiiofthecircle. Whattypeoftriangleis GKH? Arethereanyothertrianglesofthistypeinthediagram?If so,namethem. 3.Giventhatthemis113.2 Whatism?Lookbackatproblems1and2) 4.Given,whichangleswouldhavethesamemeasure? Listthemallbelowandsayhowyouknowtheyareequal. 5.Thereareseveraltrianglesinthecircle.Listthetriangles thatareinscribedtriangles. Also,listanyothertrianglesandclassifyasmanyofthe trianglesasyoucan. 6.Giventhatm GFE=70 findallpossibleangleand arcmeasurementsthatyoucan. Name: 2013www.flickr.com/photos/andy4li 29
5 Circles:)a)Geometric)Perspective)) 7.5) Giventhesimilarshapesbelowprovidethedesiredmissingsidesorproportions. 7. a. =?? Fillintheproportionandstate howyouknowitiscorrect. b. =? Fillintheproportionandstate howyouknowitiscorrect. c.ifpossible,fillinthemissingproportionssotheyaretruestatements.ifnotpossiblesaywhynot. i) =?? ii) =?? iii)? =? iv)? =? v)? =? Set Topic:Connectingpolygonswithcircles 8.Belowyouaregivenacircleandalsoseveralsquaresthatareconstructedsothattheirsidesare equaltotheradiusofthecircle.usethesesquaresandcircletoestimatehowmanysquaresittakes tofillintheareaofthecircle.statewhatyounotice.youarewelcometousetracingpaperor createcutouts.) 9.Whichofthepolygonsbelowwouldhaveanareaandperimeterclosesttothecircleitisinscribed within?why? MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
6 Circles:)a)Geometric)Perspective)) 7.5) 10.Giventhattheradiusofthecirclesinpreviousproblemis10feet.Findtheareaofeachofthe regularpolygonsandlisttheminthetablebelowalongwiththemeasureofoneangleforeach polygonandthesidelengthofeachpolygon.acouplearefilledinforyou.) Shape Oneinteriorangle Lengthofoneside Areaoffigure Triangle 60 Square 10 2=14.14 Pentagon Hexagon Octagon Circle 11.Showandexplainhowacirclecanbecutintosectorsandreconfiguredtoappearapproximately asapolygonthatcouldhaveitsareacalculatedusingastandardformula. 12.Showandexplainhowacirclecanbebrokenintoseveralringsorinteriorcirclesthatcanbe rearrangedtoappearapproximatelyasapolygonthatcouldhaveitsareacalculatedusinga standardformula. Go Topic:Findingarclengthasadistance. Just&as&a&circle&can&be&broken&into&360§ors&as&a&means&finding&the&area&of&any&size§or.&Similarly& the&circumference&of&a&circle&can&be&broken&into&360&equivalent&pieces&as&a&means&for&finding&the& distance&actually&traveled&along&any&arc&of&the&circle.& & Thecircumferenceofthecircleis6inches. So,asectorofonedegreewouldhavelengthof # inches. Andsotheareaofthesectorwithacentralangleof80 o Wouldbe80) # whichsimplifiestobe inchesor approximately4.19inches. & & MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
7 Circles:)a)Geometric)Perspective)) 7.5) Lookcloselyattheexampleonthepreviouspageandthenusethisstrategyforfindingthe arclengthactualdistancetraveledalongthepathofthearc)ineachoftheproblems providedbelow NeedAssistance?Checkouttheseadditionalresources: MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense
2n )" Tehani"drew"this"diagram"as"part"of"her"work"as"she"developed"this" formula."""
26 7.5FromPolygonstoCircles ASolidifyUnderstandingTask Part&1:&From&perimeter&to&circumference& Intheprevioustask,PlanningtheGazebo,youdevelopeda strategyforfindingtheperimeterofaregularpolygonwithn sidesinscribedinacircleofradiusr.tehani
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