# Calculate the angle between tangent lines on two points of a circle given a radius and a distance between them.

I want to create a formula that will calculate the angle change between two points on a circle, given the distance along the circumference of the circle between the two points, and the radius of the circle.

I don't really have a clue where to start.

Any help would be appreciated :)

• Start by drawing normals to the tangents connecting the points to the center of the circle. The distance along the circumference tells you the angle subtended by the normals. From there you should be able to figure out the rest. – Nimrod Jul 6 '15 at 23:29

There exists a theorem that states that the a tangent line to a circle at a point is perpendicular to the radius to that point. Since the length of the arc between those two tangent lines is known (call it $d$), the central angle is by definition of radians $r/d$. The sum of the angles in a quadrilateral is $2\pi$, and so the desired angle is simply $2\pi - \pi/2 - \pi/2 - d/r = \pi - \frac{d}{r}$.
Converting to degrees (by multiplying by $180/\pi$), the angle is then equal to: $$180 - \frac{180d}{\pi r}$$