Department of Mathematics Education

Pick's Theorem

You might have heard of Pick' Theorem. But in many cases we just use that formula without the considering why it is so. Once we know how that is made, it's hard to forget and easy to apply whenever we need.

First let me start with the follwing definitions:

**I **- Number of the interior points of a polygon on a geoboard.

**B** - Number of the boundary points of a polygon on a geoboard.

**Fundamental Triangle** - Any triangle with no interior points and
no more three boundary points

*1) How does the area change if one starts from a *fundamental
triangle(b=3, I=0), keeps B at *constant a 3 , and subsequently increases
I by one? *Write down the results of your exploration(possibly *in
a table*).

**a) To keep B at 3 point and subsequently increases I by one.**

As the interior points increases by one, the area increases by one. So we can make the below table from the pattern tha we have found.

**b) To keep B at 4 points and subsequently increases I by one.**

As the interior points increases by one, the area increases by one. So we can make the below table from the pattern tha we have found.

**c) To keep B at 5 points and subsequently increases I by one.**

As the interior points increases by one, the area increases by one. So we can make the below table from the pattern tha we have found.

Up to here we can guess that the number of interior points make the area increase by one.

Now here let me put the boundary points in the very left column and interior points in the very top row. Here is a Table for that

*2) How does the area change if one starts from a *fundamental
triangle (B=3, I=0), keeps I at *constant, and subsequently increases
B by one? *Write down the results of your exploration.

**a) To keep I at 0 point and subsequently increases B by one
**

**b) To keep I at 1 point and subsequently increases B by one
**

**c) To keep I at 2 points and subsequently increases B by one**

Up to here we can guess that the number of bondary points make the area increase by a half.

Now here let me put the boundary points in the very left column and interior points in the very top row. Here is a Table for that

Now here let make a big table to include characteristics that we found so far and lead to a Pick's Theorem.

A r e a

A(m, n) means the area of polygon when the number of boundary point is m and that of interior point is n.

So we have A(13,6) = 11.5 according to the above table.

*Step 1.*

A(13,6)

= A(12,6) + 0.5

= A(11,6) + 0.5 + 0.5

= A(10,6) + 0.5 + 0.5 + 0.5

= A(9,6) + 0.5 + 0.5 + 0.5 + 0.5

.

.

=A(3,6)+0.5+0.5+0.5+0.5+0.5+0.5+0.5+0.5+0.5+0.5

= A(3,6) + 10 * 0.5

*Step 2.*

A(3,6)

= A(3,5) + 1

= A(3,4) + 1 + 1

= A(3,3) + 1 + 1 + 1

= A(3,2) + 1 + 1 + 1 + 1

= A(3,1) + 1 + 1 + 1 + 1 + 1

= A(3,0) + 1 + 1 + 1 + 1 + 1 + 1

= A(3,0) + 6 * 1

*Step 3.*

A(13,6) = A(3,6) + 10 * 0.5

= A(3,0) + 6 * 1 + 10 * 0.5

= 0.5 + 6 * 1 + 10 * 0.5

= 0.5 + 6 + 5

= 11.5

Now we have generalization,

A(B,I) = A(3,0) + I * 1 + (B-3) * 0.5

= 0.5 + I * 1 + (B-3) * 0.5

= (B-2) * 0.5 + I

= 0.5 B + I -1

= 1/2 B + I -1