Pick's Theorem

In 1899, a man named  George Alexander Pick discovered a new and very different method of finding the area of a polygon.  This could be ANY polygon, as long as it can be placed on the Cartesian Plane and each of the vertices has coordinates that are integers.

His formula?  Let "i" be the number of lattice points (where the grid lines intersect on the coordinate plane) on the interior of the shape; let "b" be the number of boundary points (where the grid lines fall on the sides of the polygon), then Area = i + (b/2) -1.

For example, the green triangle has 4 interior points and 4 boundary points (vertices are included as boundary points).  The area is 4+(4/2)-1=5,  Another way to calculate area would help us verify this answer for area. We could verify this area found by Pick's Theorem by taking the 3x4 rectangle that completely encompasses the green triangle and subtract the three right triangle areas.
(12)  -  (1/2) 2x2 - (1/2)2x3 - (1/2)1x4 = 5. (Note: using our area estimation algorithm from class only gives an estimate, not necessarily an exact value, so don't use that method here.)

 Find the areas in TWO ways (Pick's thm and one other -- either rectifying or subtraction from a larger rectangle) shown in the previous paragraph of the remaining colored polygons.

More for practice and deeper understanding: 

Now use the illustration below to find the areas of regions A, Q and R. Verify that A = Q + R.  

Now, for your blog, find the areas outlined above using Pick's Theorem (yes, all of them) and verify them using techniques you have learned in class.  THEN, go to this website to see what happens that's interesting when you use Pick's Theorem to find areas of regions with "holes" in them.  Feel free to comment on the similarity between the proof of Pick's Theorem and how we found the areas of triangles placed on the coordinate plane in class. 

Remember your blog must be STAND ALONE. Do not require your reader to read Jammnpeaches to know what's going on in your blog. 

G. A. Pick died in a concentration camp in about 1943.  

Sources: 

http://www.cut-the-knot.org/ctk/Pick.shtml

https://en.wikipedia.org/wiki/Pick%27s_theorem

http://jwilson.coe.uga.edu/emat6680fa05/schultz/6690/pick/pick_main.htm

https://nrich.maths.org/1867