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neat, finally this account tweets a theorem I kinda intuitively get
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𝐏𝐢𝐜𝐤'𝐬 𝐓𝐡𝐞𝐨𝐫𝐞𝐦: Let P be a polygon in ℝ² whose vertices lie in the lattice ℤ². Writing Int(P) for the interior of P and ∂P for the boundary of P, we have Area(P) = |Int(P) ⋂ ℤ²| + ½|∂P ⋂ ℤ²| - 1 .
A few depicted example polygons