Interior Angles of a Polygon
Date: 05/20/97 at 19:46:32 From: Danielle Deitz Subject: Interior angles of a polygon What can you conclude about the sum of the interior angles of a pentagon? A triangle? A quadrilateral? A hexagon? Does the sum depend on whether or not the polygon is convex? Why? What would you predict the sum of the interior angles of a 20-sided polygon to be? How do you calculate this? Would the calculation be 20 x 360 because there are 360 degrees in a quadrilateral? Can you please help me?
Date: 05/21/97 at 10:59:15 From: Doctor Wilkinson Subject: Re: Interior angles of a polygon You had a good idea there to work from a simple case up to the big polygon. Unfortunately, your guess was off because you didn't see just how the smaller case relates to the big one. Let's look at the simplest cases first. For a triangle, there's an important theorem that tells you that the sum of the interior angles is 180 degrees. From this you can figure out the rest. Let's look at a rectangle next. Here you can see the answer in two ways. First, you know that all the angles of a rectangle are 90 degrees, and there are four of them, so the sum is 4 * 90 = 360 degrees. Let's look at this another way. Draw a diagaonal across the rectangle to divide it into two triangles: -------- |\ | | \ | | \ | | \ | | \ | | \| -------- The sum of the interior angles of each triangle is 180 degrees, and all the interior angles of the triagle make up the interior angles of the rectangle. Since there are two triangles in the rectangle, the sum of the interior angles of the rectangle is 2 * 180 = 360 degrees. This argument doesn't depend on having a rcctangle. Any convex quadrilateral will do. If you draw the right diagonal, you can make it work for even a non-convex quadrilateral. Now see what you can do for a pentagon. Again you can start at one vertex and draw two diagonals. This divides the pentagon into three triangles, and you can see that the sum of the angles of the pentagon is 3 * 180 = 540 degrees. Now we should begin to see a pattern. Let's make a table: number of sides sum of interior angles --------------- ---------------------- 3 1*180 4 2*180 5 3*180 .... 20 ? The part about whether it makes a difference if the polygon is convex or not is the trickiest question. The answer is no, but this is not completely obvious. The idea is that no matter how the polygon twists and turns, you can always cut off a triangle that stays entirely inside the polygon, so that what is left is a polygon with one fewer side. The general idea here, as in a lot of math problems, is to start with the simplest cases, where you can see what is happening, and work your way up to more complicated cases, looking for a general pattern. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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