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Nonconvex Polygon Angle Measure

Date: 02/03/99 at 14:07:01
From: Josh Hartman
Subject: Nonconvex polygon angle measure

What is the formula to find the interior angle measurements of a 
nonconvex polygon?

Date: 02/03/99 at 16:43:53
From: Doctor Rob
Subject: Re: Nonconvex polygon angle measure

Thanks for writing to Ask Dr. Math!

The question you ask is hard to answer for a couple of reasons. First, 
you didn't say what information is given to start with. Second, unless 
enough information is given, there may not be a unique answer. Third, 
we don't know what level of mathematics you can understand. Probably 
trigonometry and/or analytic geometry will be involved.

If you know the coordinates of all the vertices of the polygon, you can 
find the slopes of all the sides pretty easily. Then the arctangent of 
each slope will give you the angle that that side makes with a 
horizontal line, called its inclination. By subtracting the two 
inclinations of sides meeting at a vertex, you can find the interior 
angle, provided you know whether it is greater or less than 180 

Example: If the vertices are (0,0), (2,2), (5,1), and (2,7), proceeding 
cyclically around the polygon, then the slopes of the sides are 1, 
-1/3, -2, and 7/2, in order. Then the inclinations are 45, 161.565, 
116.565, and 74.055 degrees, respectively, and the angle differences 

   45 - 161.565 = -116.565 degrees
   161.565 - 116.565 = 45 degrees
   116.565 - 74.055 = 42.510 degrees
   74.055 - 45 = 29.055 degrees

Now these differ from the true angles by a multiple of 180 degrees, and 
the true angles are 243.435, 45, 42.510, and 29.055 degrees, which 
properly add up to 360 degrees. The interior angles will all be between 
0 and 180 degrees except where you have a reflex angle. If you know in 
advance which points these are, then it is a simple matter to add 
multiples of 180 degrees until the angle there is between 180 and 360 
degrees. In this case, (2,2) was a vertex at which there was a reflex 

If you don't know which points have reflex angles, the problem is a bit 
harder, as you have to identify them. One way to do that is to use the 
formula for the area of the polygon (click on Polygons) given at   

Compute the area of the polygon; then compute the area of the polygon 
omitting one vertex. If the second area is larger than the first, that 
vertex has a reflex angle. If the second area is smaller than the 
first, that vertex has an acute or obtuse angle.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Coordinate Plane Geometry
High School Geometry
High School Triangles and Other Polygons

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