I have one circle C1 with known diameter d1. That circle is surrounded by N circles (C2) of unknown diameter d2 where each of C2 circles touches two neighboring C2 circles and C1 circle in one point. Have to find d2.
Something like this…
So, what have I tried so far…
I'm not mathematician and to be honest, I haven't been in touch with any by basic everyday math for almost 20 years. If I remember well from school, solution to this problem falls under trigonometry. So I tried to use triangles in many different ways and finally ended up with something like this…
I could solve this by basic trigonometric formulas if any of the sides of triangle is known, but I have only partial data and some relations. So this is my best shot.
I don't know if this is the right way to solve it or do I have enough of known values to get desired result but that's all I have this far. Maybe my approach seems apsurd for most of you doing stuff like this on daily basis, so don't blame me.
Anyway, eneded up here because I'm trying to solve practical problem using math so guess that place where mathematicians are around is good to start with. I couldn't even search the web for solutions just because I really don't know what to look for or technical terms to search for. Hope someone here may help me with formula to get d2 from first figure when N and d1 are known.
Hope I provided enough of details to explain my problem because I really don't know any other way to do this.
Best Answer
You are doing it right. Just use the Law of Cosines on $ABC$ as follows:
$$\cos\left(\frac{360}{N}\right)= \frac{2(r_1+r_2)^2 -(2r_2)^2}{2(r_1+r_2)^2}$$