Area of a Convex Polygon

Area of a Convex Polygon

The coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃), . . . , (x_n, y_n) of a convex polygon are arranged in the "determinant" below. The coordinates must be taken in counterclockwise order around the polygon, beginning and ending at the same point.

Formula: Area = (1/2)|matrix of (x1,y1),(x2,y2),...,(xn,yn),(x1,y1)| = (1/2)[(x1y2+x2y3+...+xny1)-(y1x2+y2x3+...+ynx1)]

Example:

Find the area of this polygon: A convex polygon with vertices at (-4, 3), (2, 5), and (5, 1) plotted on a coordinate plane, shaded yellow.

Area = 1/2 times determinant with columns (2,−4,5,2) and (5,3,1,5) = 1/2[(6+−4+25)−(−20+15+2)] = 15