# Concave Polygons and Exterior/Interior Angles

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### raylee821

Jul 16, 2005, 12:09:59 AM7/16/05
to

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.

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### Kevin Karplus

Jul 16, 2005, 9:48:41 AM7/16/05
to

On 2005-07-16, raylee821 <eterna...@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 kar...@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.

### ticbol

Jul 16, 2005, 9:48:42 AM7/16/05
to

What do you mean by generalization?

As you have said, ".....the sum of the exterior angles for a convex
polygon is 360 degrees", which is true, as long as you measure the
angles in one direction only. Either all clockwise, or all
counterclockwise.

Is by generalization you mean proof?
There must plenty of proofs for this in the internet.

One could be, in a convex polygon of n number of sides---an
n-gon---there are also n number of straight lines if all the sides are
extended on one end either clockwise or counterclokwise. Each of these
straight lines is an angle of 180 degrees at the point of extension.
Hence, there are a total of n number of 180 degrees fo the whole
n-gon---in one direction only.
We know that the sum of all of the interior angles of the n-gon is
(n-2)(180deg).
Then, n(180deg) minus (n-2)(180deg) equals the sum of all the exterior
angles.

n(180) -(n-2)(180)
= n(180) -n(180) +2(180)
= 2(180)
= 360 degrees.

------------------
By the way, whether the closed n-gon is convex or concave, the sum of
all its interior angles is always (n-2)(180) degrees, and the sum of
all its exterior angles in one direction is always 360 degrees.

### raylee821

Jul 16, 2005, 3:18:24 PM7/16/05
to

Thanks for all of the great responses.

Thank you Kevin for noticing the mistake. I did mean a generalization
for the exterior angles of a concave polygon.

These were all great responses, but the reason I am asking is that I am
currently working with a company that is developing curriculum for
middle/high school students who need to review the basics before
jumping into algebra. A part of the curriculum covers basic geometry
the students should know, but never quite understood or learned. The
point is that this isn't supposed to be a full blown geometry course.

We are trying to state that for any polygon, the sum of the exterior
angles is always 360. But we will state this without discussing the
difference between a concave or convex polygon.(Because of the limited
amount of space.)

At this stage in the program, students have not yet covered negative
numbers, so the concept of considering the exterior angles to be
negative at concave vertices would seem very foreign to most of them.
For that reason, we are afraid of saying this generalization applies to
ANY polygon because we will only supply examples of convex polygons.

Does everyone agree that we should just distinguish between concave and
convex polygons and THEN talk about the sum of the exterior angles of a
concave polygon, and leave the explanation of how it applies to convex
polygons to their future geometry teachers?

Thanks again for the help and my apologies for the run-on sentences.

### Kevin Karplus

Jul 17, 2005, 2:48:12 PM7/17/05
to

On 2005-07-16, raylee821 <eterna...@yahoo.com> wrote:
> These were all great responses, but the reason I am asking is that I am
> currently working with a company that is developing curriculum for
> middle/high school students who need to review the basics before
> jumping into algebra. A part of the curriculum covers basic geometry
> the students should know, but never quite understood or learned. The
> point is that this isn't supposed to be a full blown geometry course.
>
> We are trying to state that for any polygon, the sum of the exterior
> angles is always 360. But we will state this without discussing the
> difference between a concave or convex polygon.(Because of the limited
> amount of space.)

I can see omitting concave polygons from the discussion here.

> At this stage in the program, students have not yet covered negative
> numbers, so the concept of considering the exterior angles to be
> negative at concave vertices would seem very foreign to most of them.
> For that reason, we are afraid of saying this generalization applies to
> ANY polygon because we will only supply examples of convex polygons.

I can't understand trying to cover the sum of exterior angles before
covering negative numbers---negative numbers are a much more basic and
important concept. I started negative numbers with my son in
kindergarten, only a year or so after he was counting reliably.

> Does everyone agree that we should just distinguish between concave and
> convex polygons and THEN talk about the sum of the exterior angles of a
> concave polygon, and leave the explanation of how it applies to convex
> polygons to their future geometry teachers?

(I assume you meant *concave* again---I hope that your book is more
careful about the words, as most copy editors would not notice that
error.) I still think that turtle graphics are the right way to
present the concept that the sum of exterior angles is 360 degrees to
kids who are not yet ready for formal proofs. It is easily
illustrated visually and by having the kids walk around a polygon.
The idea of positive and negative angles can be turned into left and
right turns (as in turtle logo). With this approach, the concept
students have learned to measure angles).

Starting from the much more difficult concept about the sum of the
interior angles (as one poster suggested) seems backwards to me.

------------------------------------------------------------
Kevin Karplus kar...@soe.ucsc.edu http://www.soe.ucsc.edu/~karplus
Professor of Biomolecular Engineering, University of California, Santa Cruz