Perhaps a rather elementary question, but I simply couldn't figure out the calculations on this one. Say one takes a circle centeblack at the origin with radius R. He or she then proceeds to place N circles with radius r ( R &gt; r) on the larger circles circumference equidistantly, so every 2 π / N in the angular sense. What is then the relationship between R and r such that all neighboring circles exactly touch?I've been trying to write down some equations with arc lengths and such for N = 4, but I can't seem to get anything sensible out of it.

Question

Perhaps a rather elementary question, but I simply couldn&#039;t figure out the calculations on this one. Say one takes a circle centeblack at the origin with radius   R. He or she then proceeds to place   N circles with radius   r (  R  &amp;gt;  r) on the larger circles circumference equidistantly, so every   2  π      /    N in the angular sense. What is then the relationship between   R and   r such that all neighboring circles exactly touch?I&#039;ve been trying to write down some equations with arc lengths and such for   N  =  4, but I can&#039;t seem to get anything sensible out of it.

Addison Herman · Accepted Answer

This is an answer to the similar question: what if the outer circles are placed on the boundary of the inner circle?I still find it easier to think about the inverse problem. Keeping up with the idea as before (in my earlier answer), the four centers of the outer circles will form a square. The inner circle is the unique circle going through the centers of these four outer circles. A quick drawing reveals that the square formed by the four outer circles&#039; centers is inscribed within the inner circle.Therefore the diagonal of the square is a diameter of the inner circle. The diagonal of the square has length   2  r      2  , and so the diameter of the inner circle is   2  r      2  . Thus the radius   R of the inner circle is   r      2  , and we see the relationship       R    r    =      2  .

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