To calculate the Area of a Polygon (A) using the Shoelace Formula:
\[ A = \left| \frac{1}{2} \left( \sum_{i=1}^{n} (x_i \cdot y_{i+1}) - \sum_{i=1}^{n} (y_i \cdot x_{i+1}) \right) \right| \]
Where:
The Shoelace Formula is a mathematical algorithm that provides a simple way to find the area of a polygon when the coordinates of its vertices are known. It is particularly useful because it does not require breaking the polygon into triangles and summing their areas; instead, it uses a direct calculation that involves the coordinates of the vertices in a specific order.
Let's assume the following coordinates for a polygon with four vertices:
Using the formula:
Sum1 = \(2 \cdot 5 + 4 \cdot 8 + 7 \cdot 3 + 6 \cdot 1 = 10 + 32 + 21 + 6 = 69\)
Sum2 = \(1 \cdot 4 + 5 \cdot 7 + 8 \cdot 6 + 3 \cdot 2 = 4 + 35 + 48 + 6 = 93\)
Area = \(\left| \frac{69 - 93}{2} \right| = \left| \frac{-24}{2} \right| = 12\)
The Area of the Polygon is 12 square units.