# If I have an $n$-gon, for odd $n\geq5$ with all angles equal to the angles of the regular $n$-gon, is my $n$-gon necessarily regular?

To paraphrase the title, is there such thing as an odd-sided (with at least 5 equal sides) non-regular polygon with all equal interior angles?

So for example, if I have a pentagon with all angles being $108^\circ$, then is it necessarily a regular pentagon?

I have constructed an example that I think works, but I just want to be sure.

The example I constructed is below, where you cut the red strip away from the regular pentagon and glue the halves together. The mid-line of the strip is concurrent to the line connecting the vertex to the mid-point of the opposite side.

• Sorry, I have just added conditions on $n$ to avoid that, as I forgot to do so initially. Commented Nov 5, 2015 at 11:52
• What example did you construct? Commented Nov 5, 2015 at 11:53
• hexagons can have external angles all $60$ degrees but different side lengths. Commented Nov 5, 2015 at 11:54
• I observed that the result was trivial for even $n$, which was why I had $n$ being odd in the title. Commented Nov 5, 2015 at 11:57
• Your construction is fine, but one can do more: translating any side of the $n$-gon parallel to itself does not change the angles. Commented Nov 5, 2015 at 12:00