# calculate radius of two circle given the distance between their radius??

Given two circles $$A$$ and $$B$$ with distance from their radius $$d$$ , is it possible to calculate the radius of the circles? If yes, how?

What if we add a $$3^{rd}$$ circle $$C$$ , with distance $$d_2$$ from center of $$B$$ and $$C$$ and given that $$C$$ is half of $$A$$ i.e $$r_a = 2r_c$$ , is it mathematically possible?

• This is not clear. In the first diagram, are you saying that the distance from $A$ to $B$, presumably the two centers, is $d$? If so, then that is not enough to determine the two radii. All you know is that $r_A+r_B=d$. – lulu Aug 21 '20 at 19:42
• In first case, you cannot. You can draw many circles with different radii meeting the distance d condition. In 2nd case, you can find it. – Math Lover Aug 21 '20 at 19:50
• thanks @MathLover and lulu , i suspected that the first one was lacking information... – Akash Jain Aug 21 '20 at 20:14

In the first example, the only restriction you have is $$r_A + r_B = d$$. There are infinitely many possibilities for the two radii so you cannot determine them
In the second example, notice that $$d = r_A + r_B = 2r_C + r_B$$. Therefore, $$d - d_2 = r_C$$.
Now, that you know $$r_C$$, you can compute $$r_A = 2r_c$$ and $$r_B = d_2 - r_C$$.