From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Formulas Date: 21 Apr 1998 03:00:43 GMT In article <353AD0AD.738@visi.net>, Michael R. Wezensky wrote: >I need formulas for figuring the length of sides of Pentagons, Hexagons, >and Octagons when you know the measurment from side to side or width of >the shape. If anyone could give these to me I would appreciate it. I don't usually do these questions but I just finished working these out to help my daughter through school geometry, so here you go (I hope I haven't made any transcription errors): Given a regular polygon with N sides inscribed in a circle of radius R, we are interested in the length S of a side, the length P of the perimeter, the length A of the apothem (a line segment joining the circle's center to the midpoint of a side) and the area X. Of course we have P = N * S and as is apparently not well known, we also have X = (1/2) * P * A. The angle between the line segments from the center to two consecutive vertices is clearly 2*Pi/N radians (360/N degrees), so the angle to the apothem is half that, Pi/N radians. From a description of the the basic trigonometric functions we then have A = R cos(Pi/N) and, by doubling the length of the base of a right triangle there, S = R * 2 sin(Pi/N). From this follows as above P = R * 2 N sin(Pi/N) X = R^2 * N sin(Pi/N) cos(Pi/N) This last may be rewritten with a trigonometric identity as X = R^2 * (N/2) sin(2 Pi/N). For certain special values of N theses trigonometric expressions may be replaced by formulas involving only integers and square roots. Here are the small ones. (You need a fixed-width font to read this table. TAB=8. I'll write v(x) for the square root of x; you can draw the little symbol by hand if you print this out!) N A/R S/R P/R X/(R^2) 3 (1/2) v(3) 3 v(3) (3/4) v(3) 4 v(1/2) v(2) 4 v(2) 2 5 (1/4)v(6+2v(5)) (1/2)v(10-2v5) (5/2)v(10-2v5) (5/8)v(10+2v(5)) 6 (1/2)v(3) 1 6 (3/2)v(3) 8 (1/2)v(2+v(2)) v(2-v(2)) 8v(2-v(2)) 2v(2) N cos(Pi/N) 2 sin(Pi/N) 2N sin(Pi/N) (N/2) sin(2 Pi/N) circle 1 0 2 Pi Pi Of course if you're given one of A, S, P, or X you can compute R from the table and then get the other quantities. Other values of N with "nice" expressions for 2*cos(Pi/N) include N=10: (1 + v(5) )/2 N=12: v(3) N=15: (1/4) ( 1 + v(5) + v(30-6v(5)) ) N=16: v( 2 + v(2)) N=17: it's the last term in this sequence: e1 = (17-v(17))/2 e2 = (-1+v(17))/2 e3 = (17+3v(17))/2 e4 = (-3-v(17))/2 e5 = (e2+v(e1))/2 e6 = (e3+e4v(e1))/2 e7 = (e5+v(e6))/2 (The fact that I called this last one "nice" probably has something to do with the reason my daughter doesn't like me to help her with her homework...) dave