# How to calculate the area of a polygon

The signed area can be computed in linear time by a simple sum. The key
formula is this: If the coordinates of vertex `v_i` are
`x_i` and `y_i`, twice the signed area of a polygon is given
by:

2 A( P ) = sum_{i = 0}^{n - 1} (x_i y_{i + 1} - y_i x_{i + 1})

Here `n` is the number of vertices of the polygon. A rearrangement of
terms in this equation can save multiplications and operate on coordinate
differences, and so may be both faster and more accurate:

2 A( P ) = sum_{i = 0}^{n - 1} ((x_i + x_{i + 1}) (y_{i + 1} - y_i))

To find the area of a planar polygon not in the x-y plane, use:

2 A(P) = abs(N . (sum_{i = 0}^{n - 1} (v_i × v_{i + 1})))

where `N` is a unit vector normal to the plane. The '.' represents
the dot product operator, the '×' represents the cross product operator,
and `abs()`

is the absolute value function.

Original resource: | The Delphi Pool |
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Author: | Unknown |

Added: | 2013-04-09 |

Last updated: | 2013-04-09 |