The area of the given rectangle is 6 square units.\par For $k=1$ to 5 the numbers of interior $k$ points are: 2, 15, 40, 77, 126\\ and the numbers of $k$-points on the boundary are: 10, 20, 30, 40, 50\\ giving the total numbers of $k$-points: 12, 35, 70, 117, 176.\par Assuming the quadratic formula $Ak^2 - Bk + C$ is valid for the number of interior $k$-points: \begin{eqnarray*} 2 &= A - B + C \\ 15 &= 4A - 2B + C \\ 40 &= 9A - 3B + C \\ 77 &= 16A - 4B +C \\ 126 &= 25A - 5B +C. \end{eqnarray*} Hence $A= 6, B=5, C=1$ satisfies all these equations.\par Assuming the quadratic formula $Ak^2 + Bk + C$ is valid for the number of $k$-points in the closed polygon, but not assuming that $A, B$ and $C$ take the same values:\par \begin{eqnarray*} 12 &= A + B + C \\ 35 &= 4A + 2B + C \\ 70 &= 9A + 3B + C \\ 117 &= 16A + 4B +C \\ 176 &= 25A + 5B +C. \end{eqnarray*} Hence $A= 6, B=5, C=1$ satisfies all these equations which turn out to be the same values of $A, B$ and $C$ where $A$ is the area of the polygon and $2Bk$ is the number of $k$-points on the perimeter.\par For very large $k$, where the mesh is divided into squares of area ${1\over k^2}$, the total area inside the polygon is approximately given by the number of interior points times ${1\over k^2}$ and, taking the limit as $k\to \infty$ $${\rm Area of polygon} = \lim_{k\to infty}{\rm number of interior} k-{\rm points}\times {1\over k^2} = A.$$