Summary: This paper presents a formula for the distinct dissections by diagonals of a regular $n$-gon modulo the action of the dihedral group. This counting includes dissection with intersecting or non-intersecting diagonals. We utilize a corollary of the Cauchy-Frobenius theorem, which involves counting of cycles. We also give an explicit formula for the prime number case. We give as a remark the number of distinct dissections, modulo the action of the cyclic group of finite order.

05C25, 05C69

n-gon, dissection, dihedral group