Nice work, ong.

The method of recognizing that the triangle in ong's picture is isosceles is hidden in his explanation.

When you add a line from the center of the circle to the midpoint of c, you create a pair of congruent right triangles.
Now, we have bisected the angle a, so we'll keep that in mind.

We can now write out a sine equation for the angle a/2.
sin(a/2) = {opposite}/{hypoteneus}
sin(a/2) = (c/2)/R
sin(a/2) = c/(2R)
a/2 = arcsin(c/(2R))

a = 2arcsin(c/(2R))

as ong said.

and once you know the angle a, then the arc length is R*angle (where angle is measured in radians).

So the arc length is
R*a = 2R*arcsin(c/(2R))