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Do External Angles of Polygons Always Sum to 360 Degrees?

Date: 02/13/2007 at 23:12:17
From: Cole
Subject: External angles of polygons

Please explain how a polygon with 16 right angles can have a total 
external angle of 360 degrees?

I know the rule that says that the sum of all external angles of all
polygons = 360 degrees, but I don't understand how it can work on all
polygons when some polygons can have an infinite number of right
angles which would make the sum of all external angles much more than
360 degrees.




Date: 02/14/2007 at 12:46:40
From: Doctor Peterson
Subject: Re: External angles of polygons

Hi, Cole.

A polygon can't have an _infinite_ number of angles, but I think I 
know what you mean; it can have as many right angles as you want, 
and thus can have more than enough to invalidate the claim.

The trick is that in order for that to happen, it has to be a 
concave polygon, with some of the angles bending outward rather than 
inward as usual (for a convex polygon).  And such reversed angles 
have to be counted as negative.

Let's take a simple example:

     A-----B
     |     |
     |     |
     |     C-----D
     |           |
     |           |
     F-----------E

Here we have six right angles, which, as you suggest, add up to 6*90 
= 540 degrees rather than 360.

But look at the actual external angles (I'm going clockwise starting 
at A):

     |90
     A-----B---
     |     |90
     |     |
     |     C-----D---
     |     |-90  |90
   90|           |
  ---F-----------E
               90|

One angle bends the opposite way from the others, and is taken as 
negative.  Then the total angle is 90-90+90+90+90+90 = 360.

The claim you are challenging is true for all convex polygons without 
reservation; when applied to concave polygons, you have to interpret 
it in this way, with signed angles.

See this page for another example:

  Sum of Interior and Exterior Angles
    http://mathforum.org/library/drmath/view/62228.html 

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Triangles and Other Polygons

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