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## 'Pick' printed from http://nrich.maths.org/

Investigate polygons, like those in the diagrams, with all the
vertices on the lattice points of a grid. For each polygon, work
out the area A, count the number B of grid points on the boundary,
and count the number I of grid points in the interior of the
polygon. Can you find a formula connecting A, B and I? Display your
results in a table, for example:

The following method may help you to find a formula if you do
not spot the pattern. First divide your polygon into triangles each
of which has an area of one half a square unit. Next consider the
total sum of all the angles in all the triangles in two different
ways. If you assume that any polygon can be split into triangles in
this way, then this method gives a proof of a general formula
connecting A, B and I.