Quandaries and Queries


Grade 9-10
Given only the length of an arc (72) & the length of its cord (71), how to find the radius?


Hi Robert,

Unfortunately there isn't a nice answer to your question. I drew a diagram and labeled various parts.

A is the midpoint of the chord so |AB| = 71/2. I let theta be the angle at the center of the circle subtended by the arc. Theta is in radians. Triangle ACB is a right triangle so

sin( theta/2) = 71/2r

The length of the arc subtended by the angle theta is 72 so

72 = r theta


theta = 72/r

Hence, substituting into the first equation and simplifying I get

 71/2 = r sin( 36/r)

The best you can do now is use some numerical method to approximate r.