Angles of StarsDate: 08/18/97 at 01:07:22 From: Special Ed Subject: Practical geometry What are the interior and external angles of stars built on the following regular polygons: a pentagon and octagon. I don't need lengths, just the angles. Or how about just the interior angles of a regular pentagon and an octagon? I have found definitions, but not the angles. Thank you for helping a frustrated parent. Where else should I have looked? Date: 08/22/97 at 14:43:06 From: Doctor Rob Subject: Re: Practical geometry To answer the last part first, the interior angles of a regular pentagon have measure 108 degrees, and those of a regular octagon have 135 degrees. To build a star on a regular n-sided polygon, you would add n congruent isosceles triangles whose bases are the same length as the sides of the polygon, by joining each side of the polygon to the base of one of the isosceles triangles. There is some freedom here, in that you can make the other sides of all the isosceles triangles have length equal to anything you like, as long as it is more than half the side length of the polygon. For example, you could make all the triangles equilateral. I think this option looks poor for a pentagon. The usual five-pointed star is formed by extending the sides of the pentagon until they meet outside the pentagon. In this case, the base angles of the triangles have measure 72 degrees, and the vertex angle has measure 36 degrees. When you erase the original pentagon, the large interior angles are 252 degrees, and the exterior angles are 108 degrees. In the case of an octagon, one can extend the diagonals of the octagon until they meet, in which case the vertex angle of the points has measure 45 degrees, and the base angles 67.5 degrees each. When you erase the original octagon, the large interior angles are 270 degrees, and the exterior angles are 90 degrees. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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