Pick's Theorem and area measurement: In 1899, Georg Pick discovered a remarkable method of easily calculating the area of irregular polygons. Consider the two examples shown below.
What we do is count the number of corners of a polygon and the number of dots on the unit lattice that it encloses.
The figure on the left has six corners and encloses seventeen dots. By Pick's Theorem its area is:
A = 17 + 6 / 2 - 1 = 19
The figure on the right has seven corners and encloses thirteen dots, so its area is:
A = 13 + 7 / 2 - 1 = 15.5
In general the area of any figure is :
A = enclosed dots + corners / 2 - 1
Note that if one of the edges passes straight through a dot then you count that dot as an extra 'corner' - think of it as a corner with an angle of 180 degrees. There are no such 'corners' on the two examples above.
If you've not come across Pick's Theorem before, then it's an interesting exercise to attempt to discover a proof. The theorem turns out to have important applications in number theory and other mathematics - it's not just useful for measuring areas.
For the method to work, the corners of the polygon have to be located on dots - this is equivalent to saying that the coordinates of the corners must be expressible as integers. If the corners don't have integer coordinates then it may be possible to choose a different scale where they do. If such a scale is not possible we can always choose a fine enough scale so that the area of the polygon can be approximated to any desired accuracy.