Geometry Polygons, polyhedra and solid geometry Theorem

Shoelace formula

Statement

Let $P$ be a simple polygon in $\R^2$ with vertices $(x_1, y_1), \ldots, (x_n, y_n)$ listed in order along its boundary. Put $(x_{n+1}, y_{n+1}) = (x_1, y_1)$ . Then the area $A$ of the region bounded by $P$ is $A = \frac{1}{2} \abs{\sum_{i=1}^{n} \left(x_i y_{i+1} - x_{i+1} y_i\right)}.$

Remarks

Without the absolute value, the sum gives the signed area, which is positive when the vertices run counterclockwise. Each term $\frac{1}{2}(x_i y_{i+1} - x_{i+1} y_i)$ is the signed area of the triangle with vertices $0$ , $P_i$ , $P_{i+1}$ , and these signed areas add up to the area of $P$ . Equivalently, the formula is Green's theorem applied to the line integral $\frac{1}{2}\oint (x \, dy - y \, dx)$ along the boundary. Example: for the triangle $(0, 0), (4, 0), (0, 3)$ the sum is $0 + 12 + 0 = 12$ , so $A = 6$ . Simplicity is needed: for a self-crossing figure eight the two lobes have opposite orientation, and the sum gives the difference of their areas.

Going deeper

For a student

The formula is the standard way to compute polygon areas in computational geometry and surveying. It is a discrete case of Green's theorem, and it is also the main tool in many proofs of Pick's theorem.

Reading path

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Polygon concept
Shoelace formula statement

Tag	`SDHZRWMSTT`
URI	`m2://SDHZRWMSTT`
Kind	statement, theorem
Writing	W0 stub
MSC	51M25, 51N20
Status	This copy of the site has no build data. Run `m2 status SDHZRWMSTT`.
About	Polygon
Uses	Polygon
Same as	Wikidata Q3748369 Wikipedia Shoelace formula