At the beginning of his lectures on Physics, Feynman introduces a method for measuring nuclear cross sections that is very similar to the Monte Carlo approximation of π: Take a thin slab of some material with area \(A\) and randomly shoot \(N\) high-energy particles at it. Each particle enters the material and exits on the other side if it doesn't hit a nucleus, in which case it will be stopped or deflected. If we know the number \(n\) of atoms in our material, the following equation should hold approximately for large enough \(N.\) $$ \frac{N_c}{N} = \frac{n \pi r^2}{A}, $$ where \(r\) is the radius of a nucleus (thus, assuming the nucleus is a sphere, \(\pi r^2\) is the area of a single nucleus cross section) and \(N_c\) is the number of collisions, that is, the total number of random shots minus the number of particles that went through the material undeflected. So the area of an atomic nucleus' cross section is roughly $$ A_n = \pi r^2 = \frac{N_c A}{N n}. $$
We make the simplifying assumption that all atoms are squeezed into one flat layer. In reality there will be many layers, no matter how thin we try to cut the material, but since the nuclei are so small, it's unlikely that they will overlap.