parmenides51 wrote: Is it possible to locate in a rectangle of $5$ cm by $ 8$ cm, $51$ circles of diameter $ 1$ cm, so that they don't overlap? Could it be possible for more than $40$ circles ? It is impossible to locate more than $40$ circles with diameter of $1$. See below.

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Circles in Rectangle.pdf (81kb)

For the first part, a circle with diameter $1$ has area $\frac{\pi}{4}$, so $51$ circles with diameter $1$ have area $\frac{51\pi}{4} > 40$, and the answer is hence No.