How many lines of symmetry does a regular octagon have?
I can do this problem by drawing a picture and lines of symmetry. My question about this problem is what if it is not an octagon, but any regular polygon. What is a simple way to solve the problem?
Problem: How many lines of symmetry does a regular octagon have?
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$\begingroup$I'd say there is 8 . Drawing an octagon is best but if you want to do without it , imagine drawing symmetric lines inbetween the lines of the Octagon or you can imagine drawing lines at the point where of the 2 lines meet.
Lines drawn inbetween lines = 4 Lines drawn where 2 points meet = 4 Total = 8
$\endgroup$ 1 $\begingroup$For n-gons there is always $n$ lines of symmetry. Draw different polygons and you will see.
$\endgroup$ $\begingroup$For n-gons there are always $2n$ symmetries in total; $n$ reflections and $n$ rotations. So in this case there are 16 symmetries in total, 8 reflections and 8 rotations.
A nice way to think about this is to consider where you can put each vertex. A symmetry is any permutation that preserves adjacency of vertices. Label the vertices 1 through to 8, then you have 8 choices for where to put the first vertex, 2 for the next and only 1 after that. Thus we have 16 symmetries.
If you want more information on this look up Dihedral groups.
$\endgroup$ $\begingroup$My answer I got was 10 because if you draw lines threw the octagon because it will explain more to you. - forth grader advice
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