We have that the arc and the angle that it subtends are proportional in a circle. In plain words, if the angle is doubled, the arclength is also doubled. We know that 2π is the total amount of radians in a circle and it corresponds to the whole circumference. Suppose that x is the fraction of the circumference that 5π/6 radians subtend. By proportionality, we have the following equation:
[tex] \frac{1circumf. }{2\pi} = \frac{x}{\frac{5\pi}{6}} [/tex].
Multiplying both sides with 5pi/6 we get:
[tex] \frac{5\pi}{6} * \frac{1}{2\pi} =x[/tex]
Hence, x=5/12
