How to find the arc measure of the arc cut by one side the of a circumscribed regular polygon?

Question

First, circumscribed means inside a circle, right? What does it exactly mean by cut by one side? A regular hexagon has side angles each $120 ^o$ so it has $240^o$ arc measure. But why is the answer $60^o$?

Answer 1

Arc measure seems to apply for a circular arc, using the swept angle, which is the arc length divided by the radius of the circle.

For a hexagon, two end points are $360^\circ/6=60^\circ$ away, which would be the angle of the arc between those points.

(Large version)

The above image shows a circle in which a regular hexagon is drawn such that it is fitting just inside. That circle circumscribes the hexagon. The sides of the hexagon cut the circle into circular arcs. For the side $BC$ the arc and sector are shown in red and the arc measure is displayed.

Answer 2

A polygon $P$ is inscribed in a circle $C$ if the vertices of $P$ lie on $C$. We say $P$ is circumscribed around $C$ if the sides of $P$ are tangent to $C$:

Either way, the angle "cut by one side" or subtended by one side at the center of the circle refers to the angle between rays from the center of $C$ to two adjacent (neighboring) vertices of $P$. (This angle is the same for a regular $n$-sided polygon inscribed in a circle and for a regular $n$-sided polygon circumscribed around a circle.)