How do you calculate the area of a polygon using the coordinates of its vertices?

To calculate the area of a polygon using its vertex coordinates, you can apply the Shoelace Theorem. For a polygon with vertices $(x_1, y_1)$, $(x_2, y_2)$, …, $(x_n, y_n)$, the area $A$ is given by the formula:

$$ A = 0.5 \cdot \left| \sum_{i=1}^{n} (x_i \cdot y_{i+1} – x_{i+1} \cdot y_i) \right| $$

Note that the summation runs from $i = 1$ to $n$, and $(x_{n+1}, y_{n+1})$ is the same as $(x_1, y_1)$ to close the loop. This formula is particularly useful when the vertices of the polygon are known and ordered in a consistent direction, either clockwise or counterclockwise.

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