The Cannonball problem states that you can make a rectangle and a pyramid with a rectangle base using $4900$ cannonballs (with radius $1$).
The question is, what is the density of the packing in the rectangle?
I think: the packing can be written as:
$$\frac{\text{total area of the cannonballs}}{\text{total area of the rectangle}}$$
Now, a rectangle is just a special type of a regular n gon, the area of such a gon is equal to:
$$\frac{ns^2\cot(\pi/n)}{4}$$
And the area of circle with radius $1$ are given by:
$$\pi$$
So:
$$\frac{\pi\cdot\text{number of cannonballs}}{\frac{1}{4}\cdot n\cdot\left(\text{number of cannonballs per side}\right)^2\cot(\pi/n)}$$
Now, I got:
$$\frac{\pi\cdot4900}{\frac{1}{4}\cdot4\cdot70^2\cot(\pi/4)}=\pi$$
But that is impossible because this means that the cannonballs are taking more space than the area of the rectangle in the first place.
What am I doing wrong?
A circle occupies $\frac \pi4$ of the square it is inscribed in. If the cannonball has radius $1$ the square has side $2$, which is the factor $4$ you are missing. You took $s=1$