Vertices of an Octagon

Date: 10/29/96 at 1:20:20
From: K.N.Krishna Murthy
Subject: How to compute vertices of an octagon

Hello Dr. Math,
  
I am so glad to find a wonderful site for maths questions. I would 
like to know the answer for the following question: 

Given an octagon of size 100 millimetres with center at (0,0), how do 
I find the coordinates of the 8 vertices? Is there a formula?

Thank you in advance,
Krishna

Date: 10/29/96 at 8:16:11
From: Doctor Jerry
Subject: Re: How to compute vertices of an octagon

Dear Krishna, 

Imagine a circle of radius 100 mm. We'll calculate the vertices 
of an octagon as points on this circle. From the definition of 
the trigonometric functions sine and cosine, if t is an angle in 
standard position (initial side along the positive x-axis and 
terminal side t radians counterclockwise from the initial side), 
then the coordinates of a point on a circle of radius a and 
with angle t are (a*cos(t), a*sin(t)).  With (100,0) as one vertex, 
the next vertex will be (100*cos(2*pi/8), 100*sin(2*pi/8)), the next 
will be (100*cos(4*pi/8), 100*sin(4*pi/8)), and so on.  All of these 
simplify quite a bit, since 2*pi/8 = pi/4 and cosine and sine of pi/4 
are well known.

I  hope this is clear.  You can see that I've assumed that you know 
trignonometry.  Please write back if necessary.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/30/96 at 6:37:50
From: K N Krishna Murthy Wipro Systems 11901
Subject: Re: How to compute vertices of an octagon

Hello Dr. Math,

I have another question based on the one you have just answered:

I have a square of size 100 millimetres and I want to inscribe an 
octagon in this square. The center of the square is (0,0). How do I 
find the coordinates of the vertices of the octagon?

Thank you again,
Krishna

Date: 10/30/96 at 8:31:19
From: Doctor Jerry
Subject: Re: How to compute vertices of an OCTAGON

Dear Krishna,

I'll assume that you want every other side of the octagon to be 
collinear with the square. Look first at the vertex V1 of the 
octagon just above the x-axis.  The angle between the line from 
the origin to V1 and the x-axis is pi/8.  Using a right triangle, 
the circle passing through the vertices of the octagon has radius d, 
where d = 50/cos(pi/8).  Now you can easily write down the vertices 
of the octagon, using a formula similar to the one given earlier:

V1 = (d*cos(pi/8),d*sin(pi/8))
V2 = (d*cos(pi/8+1*(pi/4)),d*sin(pi/8+1*(pi/4)))
V3 = (d*cos(pi/8+2*(pi/4)),d*sin(pi/8+2*(pi/4)))
and so on.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/