Henry writes: << In yesterday's (Tue, Jul 29) Wall Street Journal (!), a front page article talked about David P. Robbins's search for a generalization of Heron's & Brahmagupta's formulae for the area of polygons inscribed in circles, when given only their edge lengths. The article indicated that Dr. Robbins was using only pencil & paper, which I found a bit odd, considering that there are a number of symbolic algebra systems that I would imagine would be quite useful for this problem.
I wasn't able to access the article, but given the above information I don't see what the puzzle is. As long as a polygon inscribed in a circle is non-self-intersecting and contains the center of the circle in its interior, its area is the sum of the areas of all triangles T_E formed by each edge E and the center of the circle. Assuming radius = 1, and that E = edgelength, the area A_E of T_E is given by A_E = E * sqrt( 1 - (E/2)^2 ) / 2. (If the center of the circle is not in the interior of the polygon and/or the polygon is self-intersecting, then the polygon's area is a signed sum of these triangles' areas.) What am I missing? --Dan