Finding angle between two points on circle given cartesian coordinates of the points If I am given the coordinates of two points on a circle of radius'R', how can I find the angle between those two points, as well as the area of the arc between them(created on the circumference of the circle).  

If we take the dot product of the vectors from the center to these points then, $u\cdot v = |u||v|\cos \theta$  where theta is the angle you seek.
And since this is a circle $|u| = |v| = R$
If your circle is centered at the origin, and your points are $(x_1,y_1),(x_2,y_2)$
$\theta = \arccos \frac {x_1x_2 + y_1y_2}{R^2}$
