We can find the area A of any polygon which has its vertices on the lattice points of a rectangular grid, in terms of the number B of lattice points on the boundary and the number I of lattice points inside the polygon.
This solution was sent by Ling Xiang Ning, Allan from Raffles Institution, Singapore.
First, I divided the polygon into triangles, each with an area of half a square unit. If the number of triangles is T then the area of the polygon is T/2. All the grid points inside and on the boundary of the polygon become vertices of the triangles.
The total angle measurement in each triangle is 180 degrees so the total of the angles in all the triangles is given by 180T. This is made up of the total of the angles at B grid points on the boundary plus the total of the angles at I grid points inside the polygon.
At each grid point inside the polygon the angles of the triangles meeting at that point add up to 360 degrees. There are 4 points at the vertices of the polygon where the angles in the triangles meeting at those 4 points add up to 360 degrees. There are (B - 4) grid points on the boundary of the polygon where the angles in the triangles meeting at each point add up to 180 degrees.
180T = 360I + 360 + 180(B - 4)
Dividing this by 360 gives the formula for the area of the polygon:
A = T/2 = I + 1 + (B - 4)/2
A = I + B/2 - 1