What is the unit that measures distance around a circle?

The distance around a circle---it's circumference---or the length of some portion thereof---a segment---is measured in linear distance units such as inches, centimeters, feet, or meters.

The portion of a circle (the relative distance around) that a segment represents---the segment's circular arc---is measured in units of angular magnitude. In basic geometry and lay usage, the most common angular unit is the degree. In scientific and higher math usage, the most common unit is the radian. The circumference of a circle has an arc of 360° (360 degrees), or 2π rad (2 × pi radians). A radian therefore is equivalent to 180/π degrees. Other units of angular magnitude include the turn (360° or 2π rad), grad (aka gon; 1/400 turn), quadrant (1/4 turn), arcminute (1/60°), and arcsecond (1/60 arcminute).