WHAT IS THE AREA OF AN N-SIDED IRREGULAR POLYGON?

Transcription

1 WHT IS THE E OF N N-SIDED IEGU POYGON? The tpcal wa to measure the area of a pece of lad wth straght-le boudares s to ote the D coordates [, ] of the corers coectg eghborg les. Thus f we have a rregular three sded area havg corers at [,], [,] ad [,], the sde-legths are =sqrt(), =sqrt(), ad =sqrt(7). Oe ca the use the Hero formula to get the area- ( ) s( s )( s )( s ) where s s the sem permeter alterate, ad much easer, wa to get the area s to tae half the olute value of the vector product of the vectors represetg two of the sdes. Thus- Ths secod approach of fdg the area ca be eteded to a N sded rregular polgo whch ca alwas be broe up to N tragles. The addg up the sub-areas produces the total area. Cosder the followg schematc of a N sded rregular polgo cotag the coordate org (,) lg at some pot wth t- The sub-tragle shaded gra has the area-

2 ) ( )] ( ) [( Note that also represets half the area of a rhombus havg two of ts sdes have legth ad Net, addg all N tragles mag up the polgo produces the area- ] [ N Ths shows we ol eed the coordates of each of the N corers of the polgo to fd ts total area. It should produce correct values for both cove polgos such as a heago or for cocave polgos such as stars. et us beg b asg for the largest area a equlateral tragle ca have ad stll ft to a crcle of radus. Clearl the three vertees of ths tragle must le o the crcle at tervals. So oe ca choose the verte coordates, epressed Cartesa coordates, to be (, ), (-/, sqrt()/), ad (-/, -sqrt()/) The area wll thus be gve b- / / / / The umber sqrt()/= s a lttle surprsg sce t sas the equlateral tragle flls ol.99../π=.9..of the crcle area whch s gve b π. Cosder et the area of a heago of sde-legth s=. Here the total area s gve b s equal sosceles sub-tragles where the verte coordates of oe of these ca be- (, ), (,) ad (/, sqrt()/) The total area of the regular heago the s-

3 6{ } / / Crcumscrbg the heago wth a crcle of ut radus shows that the heago flls sqrt()/(π) or about 8% of the crcle. s alread stated earler the preset area determato method wll also wor whe part of the polgo boudar s cocave. Ths wll be the case for stars, Cosder the area of the followg fve poted star ow as the petagram- We show oe of the te equal sub-tragles whch mae up the petagram gra Ths sub-tragle has vertees at (, ), (, ), ad ( cos(π/), s(π/)) The petagram area wll thus be- { } cos( / ) s( / ) s( ) oog at the petagram geometr, usg the law of ses, ad havg the sdelegth of the surroudg petago be s=, we fd-

4 s( / ), cos( / ) ad s( /) s( / ) Thus the area of a petagram scrbed b a petago of sde-legth s= s- s( /) ( )(.899 s( / ) 8 cos( /) cos( / ) We ca chec ths aswer b otg that the petagram area ust equals the area p =(/)(/ta(π/)) of the crcumscrbg petago mus fve tmes the area of the sosceles tragle whose sdes have legths,, ad. The area of each of these sub-tragles s =(/)ta(π/). The area of the petagram thus becomes- ( ) ta( / ) ta( /) whch checs wth the earler result. terestg sdele of the petagram s that the oblque straght le dstace from oe of ts vertees to oe two awa equals eactl- s( / ) s ( / ) s( / ) You wll recogze that s ust the Golde ato alread well ow to the acet Grees. et us ed the dscusso b loog at the area of the four sded rregular polgo havg corers at (,), (-,), (,), ad(-,-) as show-

5 There are two was to evaluate the area of ths polgo. Oe ca add together the areas defed b the sub-tragles BC ad DC or oe ca subtract area DCB outsde the polgo from the large tragle DB. Usg the secod approach we fd- 7. The same result wll be produced b tag the other route.