calculating the radius of a circle if the distance between two points and the angle from the center are known
In a problem I'm working on, I have the following situation:
On a circle with an unknown radius, there are two lines from the center to the edge of the circle. The angle between these lines is known, as well as the distance between the points where the lines intersect the circle.
The question is, can the radius of the circle be determined with only this information?
$\endgroup$ 11 Answer
$\begingroup$Yes. Let $O$ be the centre of the circle, $P$ and $Q$ the points on the circumference, and $M$ the midpoint of $\overline{PQ}$; then $\triangle POM$ is a right triangle. Let $\theta=\angle POM$, and let $r=|OP|$, the radius of the circle. Then $\sin\theta=\dfrac{|PM|}r$. Since $\theta$ is half the known angle, and $|PM|$ is half the known distance, both $\sin\theta$ and $|PM|$ are also known, and we can calculate $r$.
Specifically, if $\alpha$ is the known angle, and $d$ is the known distance, then
$$r=\frac{d/2}{\sin(\alpha/2)}=\frac{d}{2\sin(\alpha/2)}\;.$$
$\endgroup$ 2
