Rich asked about the area of a polygon inscribed in a circle. The canonical reference is Robbins's 1993 paper cleverly titled "Areas of polygons inscribed in a circle", which takes Heron's formula -- that gives the area of a triangle inscribed in a circle, after all! -- and Brahmagupta's cyclic quadrilateral area formula, generalizes them to 5- and 6-gons, and makes some conjectures about the general case. Igor Pak's recent "The area of cyclic polygons: Recent progress on Robbins' conjectures", available off his web page ( http://math.mit.edu/~pak ), says just about everything there is to know now. As Rich guesses, the formalism is to give the polynomial eq'n satisfied by the side lengths and the area; its largest real root is the one you thought of originally, but its other roots give you the area when you inscribe it in smaller circles that give your polygon winding number >1, or where some edges go backwards. (In both cases you define area appropriately -- the integral over the plane of the "winding number around this point" function). And yes, the cyclic polygon is the largest-area one with given side lengths. --Michael Kleber On 7/18/07, Schroeppel, Richard <rschroe@sandia.gov> wrote:
Given a set of sides for a polygon, and suppose the polygon is inscribed in a circle. Bill's remark below, about swapping adjacent sides leaving the area unchanged, means we can freely reorder the sides, so the area is independent of the order for an inscribed polygon. What's the radius of the circle? If D = diameter, a side of length A will subtend an angle of 2 arcsin(a/D), and we need the sum arcsin (s/D) = pi as s iterates through the sides. This can be converted to a complicated algebraic equation by requiring the product of (sqrt(D^2-s^2) + i s) to have imaginary part 0. (And taking D as the largest real root.) As the number of sides is increased, this seems unlikely to be solvable in radicals, although it's pretty easy to tackle numerically. Q, repeated: If there are more than 4 sides, is the polygon of maximal area the one inscribed in a circle (and hence independent of the side ordering)? Q2: Given a set of sides, is the polygon of maximal area convex? Suppose you use various plausible definitions of area?
Rich
-----Original Message----- From: math-fun-bounces+rschroe=sandia.gov@mailman.xmission.com on behalf of Bill Gosper Sent: Tue 7/17/2007 12:05 AM To: math-fun@mailman.xmission.com Cc: rwmgosper@yahoo.com Subject: [math-fun] Re: solid conclusion(?)
This is analogous to the maximal area of a quadrilateral with sides a,b,c,d (or a,b,d,c): sqrt((c + b + a - d) (d - c + b + a) (d + c - b + a) (d + c + b - a)) ---------------------------------------------------------------------, 4 with diagonals (a d + b c) (b d + a c) (b d + a c) (c d + a b) sqrt(-----------------------) and sqrt(-----------------------). c d + a b a d + b c Whose formula is this? Brahmagupta's (cyclic case). His formula makes it clear that the area is maximized when the vertices lie on a circle. And what about arbitrary pentagons, etc? Are the maximal areas fixed w.r.t. permuting the sides? If the area is maximized when the vertices lie on a circle, we can freely swap adjacent sides, preserving the chord (and circumscribing) arc, and thus area. --rwg
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