Overlap, inscribe, and circumscribe a collection of simple geometric shapes to make a complex, composite figure. See more

What is the minimum side length of a square which can contain 5 non-overlapping unit circles?

Give your answer to 3 decimal places.

*Details:*

Notation: \( [ PQRS ] \) denotes the area of the figure \( PQRS \).

You may need to use a calculator to evaluate the area.

Two congruent squares \(ABCD\) and \(PQRS\) are positioned such that they share a common area defined by \(\Delta PQB\). The ratio of the area of \(\Delta PQB\) to the area of polygon \(AQRSPCD\) is \( \dfrac{3}{22} \).

If the side length of both squares is \(s\), then the perimeter of polygon \(AQRSPCD\) is \(\dfrac{m}{n}s\), where \(m\) and \(n\) are coprime positive integers. Find \(m+n\).

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