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### Angles of Stars

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Date: 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?
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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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Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons
Middle School Two-Dimensional Geometry

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