# Calculate the arc between two circles

Given the image below, how would you work out the position and radius of R?

Arc between two circles:

• You don't have enough info to solve for R. Imagine the bottom circles being physical discs. There is more than one size disc that could be supported between them. – turkeyhundt Jul 6 '17 at 23:47
• Please include what you've attempted in solving the question. It also would be helpful if you give a geometric description of the image so those that can't see it can still construct the diagram. (eg. Let $ABC$ be a triangle...) – Shuri2060 Jul 6 '17 at 23:48
• Let $\,d\,$ be the distance between the centers of the two circles. Then for any $\,R \ge (d-a-b)/2\,$ you can construct a circle of radius $\,R\,$ (externally) tangent to the two given circles. – dxiv Jul 6 '17 at 23:51
• I have been looking for more examples to help with working this out and have found this: Image from this site: joshuanava.biz/engineering-3/tangency.html I'm am more of a programmer than a mathematician so I am struggling to read/work out the solution given. Any help would be truly helpful – justachap Jul 7 '17 at 3:09

The example you mentioned is a simple one because R, the radius of the circle-to-be-constructed, is known. In that case, we only need to draw the red arc (centered at A with radius = R + a) and similarly the green arc. Their intersection will yield the point C which is the center of the required circle. In this setting, “the triangle inequality” requires $(R + a) + [R + b] \ge D$. That is, $R \ge (D – a – b)/2$, as mentioned by @dxiv.
If R must be a calculated result, then according to the cosine rule (applied to $\triangle ABC$), one more angle must also be known. For example, if $\alpha$ is known, then
$$(R + a)^2 + D^2 – 2(R + a) D \cos \alpha – (R + b)^2 = 0$$