   Associated Topics || Dr. Math Home || Search Dr. Math ### Sum of Interior and Exterior Angles

```Date: 02/14/2003 at 20:11:26
From: Student E
Subject: Geometry theorems

Is there a theorem for concave polygons about the sum of the interior
and the sum of the exterior angles?

I know that for convex polygons the sum of the interior angles is
(n-2)180 and the sum of the exterior angles is 360. However, my
geometry class believes that we might have proven that for concave
polygons the sum of the interior angles is still (n-2)180 but the
exterior angles' sum is 540. Do you know if this can be found in any
textbook or if it has been proven before?
```

```
Date: 02/14/2003 at 22:51:16
From: Doctor Peterson
Subject: Re: Geometry theorems

Hi, E.

The theorem you refer to for convex polygons applies to concave
polygons as well, IF you take the direction of the exterior angles
into account. If you don't, then you may get a variety of sums,
depending on how many angles turn in each direction.

Take this example, a concave hexagon with two "dents":

+----------+
\60    60/
\      /
120+    +120
/      \
/60    60\
+----------+

The sum of the interior angles is

60+60+240+60+60+240 = 720 = (6-2)180

as expected.

If you walk around the perimeter counterclockwise, you will turn
counterclockwise (making positive exterior angles) at all the acute
angles, but clockwise (making a negative exterior angle) at the two
obtuse angles, where you turn outward from the interior. The sum of
the signed exterior angles is therefore

120+120-60+120+120-60 = 360

This is always 360 because in going around a polygon you always make
one complete turn, and the sum of the exterior angles in this sense
tells how far you have turned. It is this sum that can be used to
prove the sum of the interior angles, since each interior angle is
the supplement of its exterior angle:

sum of A = sum of (180-B) = 180n - sum of B = 180n - 360 = (n-2)180

where A is each interior angle, B is each exterior angle, and n is
the number of angles. In our example, the supplement of interior
angle A=240 is B=180-240=-60. That's not how we normally think of
supplements, but it's the only way that makes sense for reflex angles
like this.

Now, if we sum the absolute values of the exterior angles, not
treating them as signed angles, we get a different answer:

120+120+60+120+120+60 = 600

You might like to play with this and see whether you can get any sum
at all by changing the number and size of the negative angles, or if
there is some restriction on possible sums.

Did you have an actual proof that the sum is 540 degrees for some
class of concave polygons, or did you just try it out for some
examples and find that to be true for those you tried? If you have a
proof, I'd like to see it, so we can figure out when it applies
(since it obviously is not true for all concave polygons).

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
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