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 <mwezensk@visi.net> 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