PROPERTIES OF REGULAR POLYGONS The simplest D close figures which ca be costructe by the cocateatio of equal legth straight lies are the regular polygos icluig the equilateral triagle, the petago, a the hexago. We wat here to quickly erive some of the geeric properties of such polygos icluig exterior a iterior vertex agles, the legth of their iagoals, a their area. Our startig poit is the followig sketch of part of oe of these polygos havig vertexes a sie-legth s- We have a sie regular close polygo whose exterior agles must at up to raias. Hece each of the exterior vertex agles equals- / This meas that the iterior vertex agle becomes- ( ) The result implies that the sum of the iterior vertex agles ca become quite large as icreases. For example a octago(=8) has the sum of the iterior agles equal to 6 ra. To calculate the area A() of a sie regular polygo we ee to just multiply the area of the gray sector show i the figure by. Oe fis the total polygo area to be- A( ) { sh / } s ta( ) s cot[ ]
For the obvious case of a square (=) oe has A()=s. For a hexago (=6) we get 3 3 A( 6) s As goes to ifiity we fi A()=r as expecte. From the geometry i figure 1 we also have- ( ) ( ) h rsi[ ] a s r cos[ ] We ext look at the legth of a iagoal lie coectig vertex with vertex +. Here we ca make use of the law of cosies applie to the followig triagle- We get s [1 cos( )] Hece- s (1 cos( ) For the case of a square we fi =s sqrt(), for a hexago =s sqrt(3), a for a octago =s sqrt[+sqrt()]. A iterestig result for occurs whe =5. There- ( / s) 1 [1 cos( / 5)] cos( ) 5 5 by usig a ouble agle formula for cos() a lookig up the value for cos(/5).
The quotiet term i this result is just the Gole Ratio fou well over two thousa years ago by the aciet Greeks. Its value to oe hure places reas- =1.618033988798988058683365638117703091798057686135867056 06818909707070189391138 Note that it is a irratioal umber a thus may be useful i the geeratio of prime umbers. For example, p=161803398879 is a prime. I playig with the equality- s 1 5 oe sees there exists a right triagle, kow as Kepler s Triagle, with sies a=sqrt(), b=1 a hypoteuse c=. So oe has the equality- 1 1 which says that 1 Aitioal irratioal umbers ca be geerate by fiig the iagoals to higher polyomials. Take the case of =7. Here we have- ( ) [1 cos( )] cos( ) s 7 7 1.8019377358083857063901890103318383637 to fifty places. We ca geeralize the above iagoal legths for ay sie regular polygo to the irratioal umber- N()= ( ) cos( ) s vali from = through ifiity. That is, the values for (/s) lie i the rage- N ( ) Every -sie regular polygo obeys the triagle rule show-
There are a ifiite umber of irratioals N() possible. Some of these allow expressio i terms of roots of itegers. I particular we fi the aitioal exact results- a- N(10)= cos(/10)=sqrt{[5+sqrt(5)]/) = 1.90113035903071387866675876868113976851 N(1)= cos(/1)=[3+sqrt(3)]/sqrt(6) =1.9318516557813657399863995779735678096780168 These irratioal umbers for the iagoal-sie-legth ratio are just as vali as is the Gole Ratio fou at =5. Note the approach towar N()=. As a fial cosieratio cosier bouig the polygo area A() betwee the two circles of area r a h, where r a h are the legth iicate i the first figure above. We ca write that- h s cot( ) r With a bit of mathematical maipulatio this simplifies to the iequality- cos( / ) Hece we have si( / ) 1 > si(/)/ a < ta(/)
As ->ifiity, both terms go to- =3.1159653589793386633837950881971693993751 but they coverge to this limit very slowly as Archimees(87-1BC) alreay fou out several thousa years ago. For a hure-thousa sie regular polygo we still have oly a six ecimal place accuracy for. Here is the iequality at =100,000-3.11598<< 3.1159757 A former colleague of mie here at the Uiversity og Floria was Dr.Karl Pohlhause of bouary layer fame. He tol me that whe he was a chil back i the late 18 hures he woul recall the value of by the memoic Drei komma Huss Verbrat. This meat 3 plus the year 115 whe the Czech religious reformer J. Huss was burt at the stake. U.H.Kurzweg Jue 30, 018 Gaiesville, Floria