Question

How many degrees will the skaters turn if they go once around a regular hexagon? A regular octagon? A regular polygon with `n` sides?

Answer 1

The sum of the measures of the exterior angles of a polygon is always 360°. For that matter, any closed figure will always account for `360^@`, exception being a triangle which is `180^@`

Explanation:

Interior and exterior angle formulas: The sum of the measures of the interior angles of a polygon with n sides is (n – 2)180. The measure of each interior angle of an equiangular n-gon is. If you count one exterior angle at each vertex, the sum of the measures of the exterior angles of a polygon is always 360°.

Hexagon : 6 sides

Sum of measure of exterior angles `= (6-2) * 180 = 720^@ = (4pi)^c`

Octogenarian : 8 sides

Sum of measure of exterior angles `= (8-2) * 180 = 1080^@ = (6pi)^c`

Polygon of n sides

Sum of measure of exterior angles `= ((n-2) * 180)^@ = ((n-2)pi)^c`