Too many circles in a triangle

Calculus Level pending

Suppose that circles of equal diameter are packed tightly in \(n\) rows inside an equlateral triangle such that there are n circles in \(n\)th row. If \(A\) is the area of the triangle and \(A_n\) is the total area occupied by the circles in \(n\) rows, find \(\displaystyle\lim_{n\to\infty}{A_n}/{A}\) (the diagram shows an example of \(n=4\) rows).

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