# A Unique Approach to Determining Areas of Any Polygon on a Grid

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## Chapter 1: Introduction to Polygon Area Calculation

In the realm of mathematics, we occasionally encounter brilliant theorems that impress us with their elegant simplicity in addressing complex challenges. One such remarkable theorem is Pick's theorem, which provides an effortlessly straightforward means to calculate the area of any polygon whose vertices are positioned on the lattice points of a grid. In this article, we will explore how this theorem functions. Named after the Austrian mathematician Georg Alexander Pick, who introduced it in 1899, this theorem has found utility across various mathematical disciplines, including combinatorial geometry and number theory.

### Section 1.1: Understanding Pick's Theorem

The formula for determining the area ( A ) of a polygon ( P ) with vertices on lattice points is beautifully simple:

A = I + frac{B}{2} - 1

Here, ( A ) represents the area of the polygon, ( I ) indicates the number of interior lattice points within the polygon, and ( B ) counts the lattice points located along the boundary, including edges and vertices. It’s important to note that Pick's theorem is applicable solely to simple polygons, which do not intersect themselves.

#### Subsection 1.1.1: A Simple Example: The Square

Consider a straightforward example: a square with vertices at (1,1), (1,5), (5,5), and (5,1). Clearly, this square has a side length of 4, resulting in an area of 16.

Using Pick's theorem, we find that there are 9 interior lattice points (I=9) and 16 boundary points (B=12). Hence, we compute the area as follows:

A = 9 + 8 − 1 = 16, confirming our previous calculation. While this was an uncomplicated case, we didn’t even need Pick's theorem to find the area of a square.

### Section 1.2: A More Complex Polygon

Now, let’s examine a more intricate polygon. According to Pick's theorem, we can derive the area, but we should verify this calculation manually to ensure its accuracy. The strategy is to decompose the polygon into rectangles and triangles.

Imagine the shape is divided as follows: The gray rectangle has an area of ( 2 times 3 = 6 ). For the blue triangle, we calculate:

text{Area}_{blue} = frac{1}{2} times base times height

For the green triangle, we compute:

text{Area}_{green} = frac{1}{2} times base times height

The red triangle shares the same area as the blue one. By summing these areas, we can confirm:

text{Total Area} = text{Area}_{gray} + text{Area}_{blue} + text{Area}_{green} + text{Area}_{red}

This total aligns with the findings from Pick's theorem, showcasing its broad applicability to all polygon orientations with a remarkably simple formula. Isn’t that fascinating?

## Chapter 2: Visual Learning

In the first video titled "Area of polygons, including in a grid," the presenter illustrates how to effectively calculate the area of polygons using various methods, emphasizing the practicality of Pick's theorem.

The second video, "How to Calculate the Area of Polygons," provides a detailed walkthrough of area calculations, reinforcing the concepts discussed in this article.