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Re: Concave Polygons and Exterior/Interior Angles
Posted:
Jul 16, 2005 9:48 AM


On 20050716, raylee821 <eternallytold@yahoo.com> wrote: > > Hi Everyone, > > I know that middle school/high school students learn that for any > polygon(convex or concave) with n sides, the sum of the interior angles > is 180(n2) and that the sum of the exterior angles for a convex > polygon is always 360. > > But is there any generalization for exterior angles of a convex > polygon? I've looked online and haven't been able to find any sources > that give a straight yes or no answer.
I assume that you mean *concave* polygons, since you already gave a rule for convex ones.
Actually, if you consider the exterior angles to be negative at concave vertices, then the sum of the exterior angles of an simple polygon is always 360 degreesyou have to make one complete turn to end up where you started. If the polygon is not simple (lines allowed to cross), then the sum of the exterior angles is any multiple of 360 degrees.
One of the simplest ways to see this is to use "turtle graphics". Imaging yourself walking along the edges. Each time you get to a vertex, you have to change your direction. The amount you change by is the "exterior angle". Since you end up where you started, facing the same way, you must have made an integral number of full turns.
 Kevin Karplus karplus@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus Professor of Biomolecular Engineering, University of California, Santa Cruz Undergraduate and Graduate Director, Bioinformatics (Senior member, IEEE) (Board of Directors, ISCB) life member (LAB, Adventure Cycling, American Youth Hostels) Effective Cycling Instructor #218ck (lapsed) Affiliations for identification only.
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