Associated Topics || Dr. Math Home || Search Dr. Math

### Triangles in a Polygon

```
Date: 06/14/97 at 19:36:10
From: Krishna
Subject: regular polygons

Dear Dr. Math,

A regular 18-sided polygon is inscribed in a circle and triangles are
formed by joining any three of the eighteen vertices.  How many obtuse
triangles are there?

I didn't get too far because an 18-gon is very large.  I tried to get
some idea by drawing an octagon, but I couldn't get too far with it
either.

Thank you
```

```
Date: 06/16/97 at 19:28:37
From: Doctor Wilkinson
Subject: Re: regular polygons

You started out with a very good idea: if the problem is too big, try
something smaller.  You ran into difficulties because an octagon is
still pretty big. When you have a good idea, push it! If you try a
pentagon, you may really be able to see what is going on.

For a pentagon, it's really easy. There are only 5 diagonals, and you
can spot the obtuse triangles right away: there are just 5 of them.

Now what's happening here? To get an obtuse triangle, the side
opposite the obtuse angle has to join two vertices with just one
vertex in between.  And the arc of the circle joining these two
vertices and passing through the vertex at the obtuse angle must be
less than half the circle.  If we got a semicircle, we would have a
right angle, and anything more gives us an acute angle.  So for the
pentagon, there's only this one possible shape for the triangle, and
we can rotate it into 5 different positions.

Now we're probably not ready to leap all the way to 18.  Let's try 6.

Now from vertex 1 to vertex 4 is half the circle, which would give us
a right angle, so again we can only go from vertex 1 to vertex 3,
which leaves just one vertex in between.  But now there are 6 vertices
altogether, so we get 6 possible obtuse triangles.

This time, if we go from vertex 1 to vertex 5, that's a right angle,
so we can't go that far.  But we can go from vertex 1 to vertex 4, and
now there are two vertices in between, so we get 2 triangles that way.
And again we have a triangle with the side opposite the obtuse angle
going from 1 to 3.  So that's 3 shapes of triangle, and we can rotate
them around to 8 positions, for 24 obtuse triangles altogether.

Do you think you're ready to do 18 now?

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search