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Topic: Concave Polygons and Exterior/Interior Angles
Replies: 2   Last Post: Jul 16, 2005 9:48 AM

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 Kevin Karplus Posts: 190 Registered: 12/6/04
Re: Concave Polygons and Exterior/Interior Angles
Posted: Jul 16, 2005 9:48 AM

On 2005-07-16, 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(n-2) 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 degrees---you 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
(Senior member, IEEE) (Board of Directors, ISCB)
life member (LAB, Adventure Cycling, American Youth Hostels)
Effective Cycling Instructor #218-ck (lapsed)
Affiliations for identification only.

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Date Subject Author
7/16/05 raylee821
7/16/05 Kevin Karplus
7/16/05 ticbol