# Given distances from an external point to 2 points on a circle and its center, what is the radius of the circle?

From a point outside a circle, I know the distance to two (or more if needed) points along the circle, as well as the distance to the center of the circle. Angles are unknown. How can I calculate the radius (or distance between the two points)?

From the image, a, b, and x are known. I also have measurements to more points along the circle if needed, say c and d.

• Along the distances $a$ and $b$, do you have any angle? Or distance between the two points on the circle? Without further information, it's impossible to know the radius: the two points may be anywhere on the circle, that only provides a lower bound and an upper bound for the radius. – StayHomeSaveLives Jan 3 '18 at 11:06
• @Jean-ClaudeArbaut no, unfortunately not. I had feared it was not possible to calculate without more information. Dang! – Daniel A. Burke Jan 3 '18 at 11:08
• There's not enough information, no matter how many distances you provide. Consider: If there are points with those distances on a particular circle, then there are just as many points with those distances on a slightly-larger circle, or even on a line between your point and the "center". – Blue Jan 3 '18 at 11:09
• Out of curiosity, where does the question come from? – StayHomeSaveLives Jan 3 '18 at 11:10
• @Jean-ClaudeArbaut a video game, actually. The context is it's a space game with large celestial bodies with POIs scattered across them. I'm writing a "routing" program of sorts, to determine shortest possible routes, etc. To do so, I basically need to form a Cartesian coordinate system identifying the POIs' locations along the surface of a sphere (planet/moon). We aren't given coordinates, so the only I thing I have to work with are reference distances. Fortunately, I was able to retrieve the radius, but now I face the problem of calculating all the coordinates from reference distances. – Daniel A. Burke Jan 3 '18 at 14:39

If you have measurements to more points like point $Y$ then the radius is: $R=Y-X$