Pick’s formula
Calculation of the area of a polygon
with vertices at the nodes of checkered paper.
Enter the area of the «simple» triangle.
$\color{NavyBlue}S \color{Black}=$
A «simple» triangle — no grid nodes neither inside
nor on sides, except for vertices.
Statement. Any polygon with vertices
in grid nodes can be triangulated,
i.e. split into «simple»triangles.
Draw a non-self-intersecting polygon.
$ S = N_1 /2 + N_2 - 1$
Unit of measure — squared parrot.
Please don't disturb.

Area of a polygon with integer coor­di­nates is $${\color{#359BE1}S} = {\color{#40b521}N_1} /2 + {\color{#E64D15}N_2} — 1,$$ where ${\color{#40b521}N_1}$ — the number of integer points on the polygon's boundary, and ${\color{#E64D15}N_2}$ — number of integer points that are inte­rior to the polygon.

Other etudes in “Areas and volumes”