A Closed-Form Probability Formula for Random Points Avoiding Vertex Neighborhoods in Regular Polygons

I investigated the probability $P$ that a uniformly random point within a regular n-gon ($n>6$) maintains a distance greater than the polygon's side length $l$ from every vertex. By partitioning the polygon into exclusion zones around each vertex and rigorously applying inclusion-exclusion principles, I established a closed-form expression about \( P=\frac{3n\tan{\frac{(n-2)\pi}{2n}}-2n\pi+12\pi-3\sqrt{3}n}{3n\tan{\frac{(n-2)\pi}{2n}}} \)The proof combines exclusion-zone geometry with careful handling of overlapping regions, and the result aligns with both analytic limits and numerical experiments.