a proof of pick's theorem.pdf

A Proof of Pick's Theorem

History of Pick's Theorem What is the theorem? But can you prove that? How can I use that in the classroom?About the project

To start the proof of Pick's Theorem, we first look at rectangles. Consider the following relationships of the rectangle.

For a L x w rectangle, the number of boundary points, or points that are on the sides of the rectangle, is 2L + 2w. We are counting the number of units, where oneunit is the space between points.

For a L x w rectangle, the number of interior points, or points that fall within the rectangle, is (L – 1)(w – 1).

By using theses two relationships, we can find the area of the rectangle using Pick's Theorem.

which is the formula for a rectangle. Since this holds, Pick’s Formula works.

Here is an example. Consider the rectangle. It has B = 2(5) + 2(6) = 22 boundary points and I = (5-1)(6-1) = 20 interior points. So the area = 22/2 + 20 -1 = 30 =(6)(5).

Now we look at right triangles, where one side is horizontal and one side is vertical, using what we just proved about rectangles. If we use the triangle to make arectangle where the triangle is half of the rectangle, Lxw, there are relationships about the number of boundary and interior points just like we found with therectangles.

If we let L be one side of the triangle (and one side of the rectangle), w be the other side, and h be the number of points on the hypotenuse of the triangle that are notendpoints of the hypotenuse, we get that the number of boundary points is

When we found the relationship for the interior points of a rectangle, we got that I = (L - 1)(w - 1). If we divide this rectangle into two equal triangles, some of theinterior points of the rectangle become boundary points of the triangle. These points fall on the hypotenuse, h. So, the number of interior points for one of thetriangles in the rectangle is

Using what we just found for the boundary points and interior points of a triangle, we can use Pick’s theorem to calculate the are of the triangle.

But can you Prove that? http://jsoles.myweb.uga.edu/proof.html

1 sur 4 19/11/2012 15:07

which is the formula for the area of a triangle.

Now we consider a triangle where only one of its sides is either vertical or horizontal. Using what we have already proved, construct a L x w rectangle around thetriangle using two of the vertices of the triangle as vertices of the rectangle.

The rectangle we have constructed can now be seen as the sum of three triangles. We also know that the area of the triangle we are trying to find (the red triangle)can be found by subtracting the area of the other two triangles by the area of the rectangle. Using the same strategy as before, we can calculate the number ofboundary and interior points. Once again, L and w mean the length of that side and H1 and H2 are the number of points on that line that are not also the vertices ofthe triangle.

The number of boundary points is

The number of interior points is equal to the number of interior points of the rectangle, minus the number of interior points of the two other triangles.

Now, using Pick’s formula, we can calculate the area of the red triangle.

which equals the area of a triangle.


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Now we consider a triangle where no sides are horizontal or vertical. Like the other triangles we will construct a rectangle around the triangle, this time using one ofthe vertices of the triangle as the vertex of the rectangle.

Once again we can find the formulas for the number of boundary and interior points of the triangle.

The number of boundary points equals

The number of interior points equal the number of interior points of the rectangle – the hypotenuses of the three right triangles and then the number of interior pointsof the three right triangles.

Now using Pick’s theorem, we can find the area of the triangle by finding the area of the rectangle and subtracting the area of the three right triangles.

If we reduce the above expression in a different way, we get that:

So the area of the triangle equals the area of the rectangle minus the area of the three right triangles.


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Now we will look at polygons. We have already proved that Pick’s Theorem works for any triangle. We will now prove that Pick’s Theorem works for any polygon.We will be able to prove this if the following two conditions are met.

1. Any polygon on a lattice can be divided into triangles. You can find a proof of this theorem here .

2. The area of any polygon is equal to the sum of its partitions. We will now prove this theorem using a polygon, P that has been divided into two partitions, Q and R.The path that is drawn to divide the polygon will include x number of points.

Since we now know the sum of the areas of the partitions of any polygon equals the area of the polygon, any polygon can be partitioned into triangles, and Pick’sTheorem is true for any triangle, we now that Pick’s Theorem will work for any polygon.

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4 sur 4 19/11/2012 15:07

a proof of pick's theorem.pdf

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