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:

A | B | I |
---|---|---|

16 | 8 | 13 |

64.5 | 13 | 59 |

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.