The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Inactive » k12.ed.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Kevin Karplus

Posts: 190
Registered: 12/6/04
Re: Concave Polygons and Exterior/Interior Angles
Posted: Jul 16, 2005 9:48 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 2005-07-16, raylee821 <> 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
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 #218-ck (lapsed)
Affiliations for identification only.

submissions: post to k12.ed.math or e-mail to
private e-mail to the k12.ed.math moderator:
newsgroup website:
newsgroup charter:

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.