Given a polygon constructed on a grid of equal-distanced grid points (i.e., points with integer coordinates):

**Pick's theorem** provides a simple formula for calculating the area, **A**, of this polygon in terms of the number, **I**, of **interior points** of the polygon and the number,
**B**, of **boundary points** on the polygon.

Formula: '''A = I + ½B − 1 .

Thus, in the above example, **I = 9**, and **B = 14**. Hence, '''A = 9 + ½(14) − 1 = 9 + 7 − 1 = 15 (square units).

This is so simple that it has been correctly used by first-grade children, drawing figures on square tiles on floor or wall, or stretching strings from pegs in pegboard. They learn how to add, along with subtraction as "take away". They learn to "halve" by a one-to-one correspondence between counters.

The formula can be generalized to three dimensions and higher by
**Ehrhart polynomials**. The formula also generalizes to surfaces of polyhedra.

### Proof

Since we are assuming the theorem for P and for T separately,

**A**_{PT}= A_{P}+ A_{T}**= I**_{P}+ ½B_{P}− 1 + I_{T}+ ½B_{T}− 1**= (I**_{P}+ I_{T}) + ½(B_{P}+ B_{T}) − 2**= I**_{PT}− (C − 2) + ½(B_{PT}+ 2(C − 2) + 2) − 2**= I**_{PT}+ ½B_{PT}− (C − 2) + ½(2(C − 2) + 2) − 2**= I**._{PT}+ ½B_{PT}− 1

So the theorem is true, if the theorem is true for a single triangle.

The verification for this case can be done in these short steps:

- observe the truth of the theorem for unit squares;
- from there see that it works for any rectangle with sides parallel to the axes;
- verify from that case that it works for right-angled triangles obtained by cutting such rectangles along a diagonal;
- now reduce the case of an acute-angled triangle case to the second case, by adding right-angled triangles to make up a rectangle;
- observe that this gives us the general parallelogram case, joining two triangles along a diagonal;
- the general triangle case follows by making two into a parallelogram.

### References

http://www.katev.org/maths/stands/AREA1.HTM

http://mathworld.wolfram.com/PicksTheorem.html