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