I’ve linked to a quick picture since this is kind of hard to ask otherwise.
What is the relationship between the length from B to A (straight line, not around the perimeter) and line BC? From rough measuering, I see that it is somewhere around 2.4 times larger. But I am sure there is a more accurate answer than that. And that it probably involves Pi.
Long Answer: Let’s say BC is length x. If we draw a line straight from C to intersect the line from AB, then that defines a 45/90/45 right triangle of which the base is BC with sides of length x/sqrt(2). Then we’ll see that the line from AB is made up of two of those lengths and one length equal to BC, so that makes that length equal to 2 * x/sqrt(2) + x.
Funny…I just stumbled on the fact that the number 2.414213562373095… must arise enough in math or nature that it is known as the silver ratio (as opposed to the golden ratio 1.6…) and is typically defined as 1 + (2)^1/2 which just happens to equal sqrt[3 + 2((2)^(1/2))] shown above.
I was constructing two different right angle triangles by extending the vertical and horizontal edges until they met. The triangle created by the 45° side of the octagon and the extended edges has hypotenuse 1, so its legs are 1/sqrt(2), so the sides of the big triangle are 1+1/sqrt(2) = (1+sqrt(2))/2 and the result follows.