Pick's Theorem Method for finding the Area of a Polygon on a dot grid

It is relatively simple to calculate the area of a rectangle. A three by three square will have an area of 9 square units calculated by multiplying 3x3. A four by five rectangle has an area of twenty square units calculated by multiplying 4x5. A 4x5 triangle has an area of 10 square units calculated by multiplying 4x5 and dividing that rectangle by 2. Essentially, if you multiply the two perpendicular (90 degree angle) dimensions, you can find the are of any two dimensional shape.

But what if the polygon is not a simply triangle or quadrilateral (four sides)? On a lattice grid, you can break up the polygon into individual rectangles and triangles. But what if there are parts of the area that do not fit a carved out triangle? This calculation is important as all images on a computer are polygons created on a microscopic lattice grid. 

Georg Pick, a mathematician, in 1899 devised a way to find the area in any lattice polygon. 

First, count all of the dots on the perimeter of the polygon and cut that number in half.

Second, count all of the dots in the interior of the polygon and add that number to half the number of dots in the perimeter.

Finally, subtract the number 1 from your result above and
that is the area of the polygon in square units.

Pick's Formula: P/2 + I -1 = Area of Polygon

• P = # of Perimeter Dots
• /2 = cut in half or divide by 2
• I = # of Interior Dots
• 1 = subtract 1 from the answer

This lesson was created to introduce the children to the concept of two dimensional area dealing with irregular polygons. I challenged the children to find the square area of several difficult polygons. Just counting the perimeter dots and interior dots was a challenge for some. I focused the on the best practice for counting: mark your first point and touch each point counting by twos until you reach your goal. The second objective was to teach them how to divide by two; with an emphasis on dividing odd integers by two. For example, if you are dividing 27 perimeter dots by two, take the largest even number in 27 and see it as 26 + 1, you can take half of 26 (13) and then half of 1 (1/2) and then add 13 and 1/2. This is the distributive property of multiplication (times by 1/2).

Of course, I strongly encourage the children to create their own complex polygons and find the area.