Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
2  
yes, isosceles triangle with known vertex angle and opposite side: law of cosines. –  Will Jagy Oct 5 '12 at 21:50

1 Answer 1

up vote 3 down vote accepted

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

$$r=\frac{d/2}{\sin(\alpha/2)}=\frac{d}{2\sin(\alpha/2)}\;.$$

share|improve this answer
    
Thank you very much, thanks to your answer I managed to solve the problem I had. –  Wouter van den Heuvel Oct 5 '12 at 22:06
    
@Wouter: You’re very welcome. –  Brian M. Scott Oct 5 '12 at 22:11

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.