A quick word on Pick's Theorem

Pick's theorem is one fun theorem which connects the discrete concept of lattice to continuous concept of Area . If you don't know the proof it seems quite inscrutable. However the idea behind proof is very simple but first the theorem. Pick theorem says that if you have a random geometrical object of course with the conditions that it has to be enclosed. Made up of straight lines, having no holes and not two different object sharing the vertex.
The area is given by L-1/2B-1 where L is the total number of lattice vertices including boundary and inside.

         The easier way to think about the proof is in terms of angles and the area of primitive triangle. The primitive triangle is any triangle where L= B = 3 i.e it is made up of only 3 vertices and its area is 1/2. If you can convince yourself that for any geometrical object it can always be triangulated by these primitive triangles then to find the area of the whole polygon all you have to do is find the number of such primitive triangles and multiply by 1/2.

  How to count the number of those primitive triangles. Well its not difficult. First notice that any interior point is surrounded by triangles and sum of all the angles is 360. So if there are I interior vertices we have 360 *I. Second if B are the vertices on boundary the sum of all interior angles of a polygon with B vertices is given by (B-2)*180. Let T be the number of triangles then the sum of all the angles of Triangles is T*180. Thus
T*180 = 360*I+(B-2)*180
which simplifies to
T=2*I+(B-2)
Now area of triangle is 1/2*B, multiplying both the sides by 1/2 we get
A =1/2*T=1/2*(2I+(B-2))
A =I+1/2B-1
Now I = L-B (Interior = Total - boundary vertices)
A = L-B+1/2B-1
A = L-1/2B-1
  proved