Associated Topics || Dr. Math Home || Search Dr. Math

### 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.
```

```
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/
```
Associated Topics:
High School Geometry
High School Symmetry/Tessellations
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search