Q.If two arcs of the same length in two circles subtend angles $65°$ and $110°$ at the centre. Find the ratio of their radii.

For a sector of a circle of radius $r$ whose central angle is $\theta$ radians, the length $S$ of the enclosed arc is directly proportional to $\theta$ (as a fraction of the full circumference $2\pi r$ corresponding to the full angle $2\pi$):

$\frac{S}{2\pi r} = \frac{\theta}{2\pi} \quad\Longrightarrow\quad S = r\theta$ …