Area of polygon

A polygon is a closed sequence of segments in the plane. It is often convenient to represent it as a sequence of vertices P₁, P₂, ..., P_n, with the convention that any pair of adjacent vertices in the sequence define a segment in the polygon, and that the first and last vertices are the same. Each point P_i is assumed to have coordinates in the plane (x_i, y_i).

Simple polygons

A simple polygon is a non-self-intersecting one, i.e. no pair of segments intersects each other.

Formula

The signed area of the polygon is given by the following formula:

The area is signed because going in reverse order means having a result with the opposite sign. To get the area it's necessary to take the absolute value.

Algorithm

The following algorithm translates the formula in pseudocode:

/*
    Vs is an array of vertices. Each vertex is supposed to be
    an array itself, with X values in index 0 and Y values in
    index 1. It is also assumed that the last element in Vs is
    the same as the first one.
*/
polygon_area_simple (Vs) {
    N = Vs.size; # number of elements in Vs
    sum = 0;
    for i in 0 .. (N - 1):
        sum = sum + Vs[i][0] * Vs[i+1][1] - Vs[i+1][0] * Vs[i][1];
    return abs(sum) / 2;
}

Performance

The algorithm for simple polygons is linear with respect to the number of vertices (O(N)).

Implementation

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Helpful links

• Polygon