Can you ever have a polygon that is regular and concave? - Answers

No. The total of the interior angles of a polygon is given by: total_of_interior_angles = 180° × (number_of_sides -2) A regular polygon has every interior angle the same, and the size of each angle is given by: size_of_interior_angle = total_of_interior_angles / number_of_sides → size_of_interior_angle = 180° × (number_of_sides -2) / number_of_sides But as number_of_sides - 2 is (always) less than number_of_sides, (number_of_sides -2) / number_of_sides is less than 1 and so the size_of_interior_angle is less than 180°. To be a concave polygon, at least one interior angle must be greater than 180°. But as a regular polygon has all interior angles less than 180°, no regular polygon can be concave.