[Math] Finding the angle of a sector

geometry

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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.

Best Answer

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).

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