When 3 or more points are joined to form a polygon on the Cartesian plane, its area can be found using the Shoelace Method.
Suppose a polygon has vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3), \dots, (x_n, y_n) \)
The area is given by:
Area \( = \frac{1}{2} \left| \begin{matrix} x_1 & x_ 2 & x_3 & \dots & x_n & x_1\\ y_1 & y_2 & y_3 & \dots & y_n &y_1 \end{matrix} \right| \)
Area \( = \frac{1}{2} \left| [x_1y_2 + x_2y_3 + x_3y_4 + \dots + x_ny_1] - [y_1x_2 + y_2x_3 +y_3x_4 + \dots + y_nx_1] \right| \)
where the polygon has \(n\) vertices
How to use the Shoelace Method:
Write the coordinates in the order that the vertices are connected, either clockwise or anti-clockwise.
Repeat the first point at the end of the table.
Find the first sum by multiplying each \(x\)-coordinate by the \(y\)-coordinate in the next row (downwards to the right).
Find the second sum by multiplying each \(y\)-coordinate by the next \(x\)-coordinate (upwards to the right).
Subtract the second sum from the first sum.
Take the absolute value (ignore any negative sign), then divide the result by \(2\).
Important Notes
The vertices must be listed in the order they are connected around the polygon.
The order can be clockwise or anti-clockwise, but do not mix the order.
Always repeat the first vertex at the end before carrying out the multiplications.
Use the absolute value to ensure the area is always positive.