# Determine a circle's radius given its intersection with another circle

Two dissimilar circles intersect at two points. The intersection points in the smaller circle are along its diameter. We know r1, the smaller circle's radius, and a, the distance between the two circles at their furthest separation. We need to solve for r2, the larger circle's radius, in terms of the other two values.

• Every circles are similar... – zoli Nov 2 '17 at 18:11
• OK, so then what I meant was, two circles of different sizes. – Paul Holmes Nov 2 '17 at 18:15
• Thanks. Thanks. – zoli Nov 2 '17 at 18:20

$a + r_1 = r_2 + \sqrt {r_2^2 - r_1^2}\\ a + r_1 - r_2 = \sqrt {r_2^2 - r_1^2}\\ a^2 + r_1^2 + r_2^2 + 2ar_1 - 2ar_2 - 2r_1r_2 = r_2^2 - r_1^2 a^2 + 2r_1^2 + 2ar_1 = 2(a + r_2)r_1\\ r_2 = \frac {a^2 + 2r_1^2 + 2ar_1}{2(a+r_1)} \\$
Draw a line from $O_2$ to $P$ (which has length $r_2$) and the use of Pythagoras becomes obvious.