Get the angle in a circle using center, radius and one point in a circle.

Question

There is a circle and i know Point1 this is fixed and i know another point Point2 which can be anywhere in the circle. and i want to know the angle which is made at center.

Thanks Your help will be appreciated.

Answer 1

You can find the angle of each radius using the Atan2 function, then subtract them to get the angle between them. If you search the site for Atan2 it comes up often. One example is here You then subtract one angle from the other. Depending on your application, you may need to add or subtract $2\pi $ to get the result in the range $[0,2\pi )$

Answer 2

Connecting the three points, we have formed a SSS triangle. That is, we can use the distance formula to get all three sides of this isosceles triangle. Now let $r$ be the radius, let $\theta$ be the desired angle, and let $c$ be the side connecting Point $1$ to Point $2$. Then using Cosine Law, we can solve for the angle to be:

$$ \theta = \arccos\left(\frac{r^2 + r^2 - c^2}{2rr}\right) = \arccos\left(\frac{2r^2 - c^2}{2r^2}\right) $$