If a circle of unit radius is divided into two parts by an arc of another circle subtending an angle $60^{o}$ on the circumference of the first circle, then the radius of the arc is

Let the centers of the two circles be O and P, with radii 1 and r respectively. The arc of the second circle subtends a 60 degree angle on the circumference of the first circle, meaning it subtends a 120 degree angle at center O of the first circle. In triangle OAP, where A is a point of intersection, angle AOP is 60 degrees and angle OAP is 90 degrees because the circles intersect orthogonally. This makes triangle OAP a 30-60-90 right triangle where the radius of the arc, AP, is opposite the 60 degree angle. Since the hypotenuse OP is 1, the side AP equals 1 multiplied by the tangent of 60 degrees, which is the square root of 3.

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