Finding the angle of a sector I have a circle, with an object lying at the edge. In the diagram the object is represented by the blue circle. I need to form a sector in the same way that is drawn in the diagram, given the following:

• The distance between the centre of the circle and the object, i.e. the radius r.
• The width of the object.
• The ratio between the width of the object and the length of the sector arc. e.g. 50%, meaning that half the arc length would be covered by the object.

I basically need to calculate the angle alpha that would satisfy the given ratio.

• Do you know object's width ? – Jean-Claude Arbaut Mar 16 '13 at 14:53
• If the object has radius $\rho$, then the angle required is $4(2 \arcsin \frac{\rho}{2r})$. – copper.hat Mar 16 '13 at 15:08
• Sorry I should have mentioned that I have the width of the object. I've fixed the question. – KkovAli Mar 16 '13 at 20:02

Let $w$ be the width of the object, $r$ the radius of the circle, $\rho$ the ratio of the width to the arc length and $\alpha$ the total angle of the arc.
The arc length is $\alpha r$, and $\rho = \frac{w}{\alpha r}$. Hence $\alpha = \frac{w}{\rho r}$ (radians).