Lines of Symmetry in Regular Polygons

Date: 03/13/2001 at 00:33:44
From: Wendy
Subject: Lines of reflectional symmetry

I am trying to find the formula for finding all the lines of 
reflectional symmetry in regular polygons. I know that you can draw 
the lines of symmetry in the shapes - they divide the figure into 
congruent halves. But my problem is I have to figure out the formula 
that could be used to solve for an n-gon.  

The shapes we were given are: pentagons, hexagons, heptagons, 
octagons, nonagons, decagons, dodecagons, and n-gons.  I have tried to 
figure this out for days but I haven't been able to find a pattern.  
Please help.

Date: 03/13/2001 at 10:49:42
From: Doctor Rob
Subject: Re: Lines of reflectional symmetry

Thanks for writing to Ask Dr. Math, Wendy.

The lines of symmetry of a regular n-gon are the bisectors of all its
interior angles and the perpendicular bisectors of its sides. There
are two cases: n even and n odd.

If n is even, there are n sides and n angles, so there are at most
2*n such lines. This counts too much, however, because every angle
bisector bisects two opposite interior angles, and every perpendicular 
bisector of a side bisects two opposite sides.

If n is odd, there are still at most 2*n such lines. Again, this 
counts too much, because every angle bisector is the perpendicular
bisector of the opposite side, and every perpendicular bisector of a
side bisects the opposite angle.

I leave it to you to finish.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/