Travis,
I am not sure I understand but let me try. My understanding is that you have an octagon as in the diagram below, you can measure the distance s and you want to calculate the distance r.
If my understanding of the problem is correct then you can proceed as follows.
If you join each vertex to the center C of the octagon then you divide the octagon into 8 congruent triangles.
The measure of the angle ACB is then ^{130}/_{8} = 45^{o}. Draw a line from C to the midpoint M of the line segment BA then ACM is a right triangle, the length of AM is ^{s}/_{2} and the measure of angle ACM is ^{45}/_{2} = 22.5^{o}.
Using trigonometry sin(ACM) = ^{AM}/_{AC}. But sin(ACM) = sin(22.5^{o}) = 0.3827 and
AC = r, thus
r = ^{AM}/_{0.3827} = ^{s}/_{(2 * 0.3827)} = ^{s}/_{0.7654 } = 1.307 s.
Penny
