Number of Lines of Symmetry in a Regular Polygon

Date: 03/12/98 at 09:48:35
From: Bonnie Cook
Subject: Lines of Symmetry in a polygon

In a regular polygon, are the lines of symmetry the same as the number 
of lines or angles of that polygon? For example, in a regular 
pentagon, are there 5 lines of symmetry? In a regular hexagon, are 
there 6 lines of symmetry? In a regular octagon, are there 8 lines of 
symmetry?

This question came up in our class discussion. I have looked in my 
math book, but cannot find an answer.

Date: 03/12/98 at 11:26:52
From: Doctor Rob
Subject: Re: Lines of Symmetry in a polygon

Every line of symmetry must be the bisector of one of the angles, or 
the perpendicular bisector of one of the sides. If a line of symmetry 
passes through a vertex, it must divide the angle into two equal parts 
(draw a picture, and prove this to yourself, using the definition of 
"line of symmetry"). If it intersects a side in a point other than a 
vertex, it must be at the midpoint (again, prove this to yourself), 
and it must be perpendicular there (same proof as the first assertion 
above).

Now sometimes, an angle bisector is a perpendicular bisector of one of 
the other sides, and vice versa (this happens if the number of sides 
is odd). Other times, an angle bisector also bisects one of the other 
angles, and the perpendicular bisector of a side is also the 
perpendicular bisector of another side (this happens if the number of 
sides is even). In either case, if you add up the number of angles and 
the number of sides, you will be counting each line of symmetry twice. 
Thus, the number of lines of symmetry is the same as the number of 
angles and also as the number of sides.

Convinced?

-Doctor Rob, The Math Forum
Check out our web site! http://mathforum.org/dr.math/