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 degrees. 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 are 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 angle. 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 http://mathforum.org/dr.math/faq/formulas/faq.analygeom_2.html 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 http://mathforum.org/dr.math/
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