G-GMD Circumference of a circle

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1 Illustative G-GMD Cicumfeece of a cicle Task π Suppose we defie to be the cicumfeece of a cicle whose diamete is : > 0 π Explai why the cicumfeece of a cicle with adius is. IM Commetay π The cicumfeece of a cicle of adius is. This well kow fomula is take up hee fom the poit of view of similaity. It is impotat to ote i this task that the defiitio of π aleady ivolves the cicumfeece of a cicle, a paticula cicle. I ode to show that the atio of cicumfeece to diamete does ot deped o the size of the cicle, a similaity agumet is equied. Two diffeet appoaches ae povided, oe usig the fact that all cicles ae simila ad a secod usig simila tiagles. This fome appoach is simple but the latte has the advatage of leadig ito a agumet fo calculatig the aea of a cicle. High school studets will kow that the cicumfeece of a cicle of adius is ad theefoe the goal of this task is to help them udestad this fomula fom the poit π

2 Illustative of view of similaity. The fist solutio equies a geeal udestadig of similaity of shapes while the secod equies kowledge of similaity specific to tiagles. A alteative agumet usig tigoometic atios povides a fomula fo the cicumfeece of a egula polygo with sides iscibed i a cicle. A egula 3 polygo with sides ca be decomposed ito isosceles tiagles by dawig lie segmets coectig the cete of the cicle to the vetices of the polygo. A example of such a tiagle (take fom a egula hexago) is pictued below: As oted i the pictue, these isosceles tiagles have two sides of legth, the adius of the cicle, ad the thid side AC has a legth b which depeds o m( ABC). The poit P i the pictue is the midpoit of AC AP. So we have = si ABP ad fom AB this we ca deduce that AP = si ABP ad the, usig that P is the midpoit of,. We kow that agles coguet to make up a full AC b = si ABP m( ABP) = 360 ABP cicle so. The peimete of the egula sided polygo iscibed i a cicle is times the side legth of this polygo, which we have just calculated: 360 si ( ). Tigoometic atios do ot deped o the size of a tiagle ad so these fomulas fo P allow us to deduce, as i Solutio, that the atio of the cicumfeece of a cicle to its diamete does ot deped o the size of the cicle. I additio, this povides a simple quatitative way to estimate π by choosig a lage value of : of couse this equies kowig the sie of a agle. This fist agumet is a example of MP7, Look Fo ad Make Use of Stuctue. The key to this agumet is idetifyig that all cicles ae simila ad the applyig the meaig of similaity to the cicumfeece. The secod agumet exemplifies MP8, Look Fo ad Expess Regulaity i Repeated Reasoig. Hee the key is to compae the cicle to

3 Illustative a moe familia shape, the tiagle. A paticulaly efficiet method is to divide a egula polygo appoximatig the cicle ito coguet tiagles, makig the fomula fo the peimete ad its elatioship to paticulaly clea. Solutios Solutio: Similaity of cicles Below is a pictue of a cicle of diamete, labelled C, ad diamete d =, labelled C : I the case pictued, d is lage tha. All cicles ae simila ad i this case the scale facto goig fom the cicle of diamete to the cicle of diamete is. The cicumfeece of a cicle is a oe dimesioal measuemet ad so it scales i the same way as the diametes: Cicumfeece( C ) = diamete( C ) Cicumfeece( C ) diamete( C ) = Sice the cicumfeece of C is π by defiitio, it follows fom the above equatio that the cicumfeece of C is π.

4 Illustative Solutio: Similaity of tiagles I this solutio we appoximate the cicumfeece of a cicle usig polygos ad the use similaity of tiagles to explai the fomula fo the cicumfeece of a cicle. Below is a pictue of a egula octago iscibed iside a cicle of adius : The cicumfeece of the cicle is a little bit moe tha the peimete of the egula octago which we ca calculated usig the pictue below: The peimete of the octago is b 8b sice it has bee divided ito eight coguet tiagles each with a base of. We ca calculate the agles of these eight tiagles usig the fact that the eight ie agles combie to make a 360 degee cicle so each measues 45 degees. The tiagles ae all isosceles so this meas that the base agles each measue = 67.5 degees. By AAA, two tiagles with agles 67.5, 67.5, ad ae simila. Theefoe the atio does ot deped o the size of the 45 (b : ) (peimete(octago) : ) egula octago. This meas that the atio also does ot deped o the size of the egula octago. As we add moe ad moe sides, this atio appoaches the atio of the cicumfeece of the cicle to its adius. We coclude that fo a cicle of ay adius C (cicumfeece(c) : ) = ( π : ).

5 Illustative Note the comes fom lookig at the cicle of diamete ad cicumfeece π: the adius of this cicle is. This is equivalet to the usual fomula sayig that the cicumfeece of a cicle with adius is. π G-GMD Cicumfeece of a cicle is licesed by Illustative ude a Ceative Commos Attibutio-NoCommecial-ShaeAlike 4.0 Iteatioal Licese

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