$$\pi \approx \lim_{k \to \infty} \frac{5^{-k}}{2} \cdot \left|\sum_{i=1}^{4(k+1)} (x_i y_{i+1} - x_{i+1} y_i)\right|$$
Derivation
This lemma states
For any non-negative integer $k$, $x^2+y^2=5^k$ has $4(k+1)$ integer solutions.
For a circle, the area is given by
$$A = \pi r^2$$
For a polygon, the area is given by the Shoelace formula
$$A = \frac{1}{2} \left| \sum_{i=1}^n (x_i y_{i+1} - x_{i+1} y_i) \right|$$
When the lattice polygon closely approximates the circle, their areas are nearly equal
$$\pi r^2 \approx \frac{1}{2} \left| \sum_{i=1}^n (x_i y_{i+1} - x_{i+1} y_i) \right|$$
Substituting $n=4(k+1)$ and $r^2 = 5^k$, then as $k \to \infty$
$$\pi \approx \lim_{k \to \infty} \frac{5^{-k}}{2} \cdot \left|\sum_{i=1}^{4(k+1)} (x_i y_{i+1} - x_{i+1} y_i)\right|$$
Example
For $k=4$, there are $20$ polygon vertices, $\pi \approx 3.0901699$
For $k=3600$, ran a C program, $\pi \approx 3.1415926$
Question
Is this a correct new method to estimate $\pi$?