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  "body": "\n## Statement\n\nLet $P$ be a simple [[polygon|8VEFWD3EJB]] in $\\R^2$ with vertices $(x_1, y_1), \\ldots, (x_n, y_n)$ listed in order along its boundary.\nPut $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.\nThen the area $A$ of the region bounded by $P$ is\n$$A = \\frac{1}{2} \\abs{\\sum_{i=1}^{n} \\left(x_i y_{i+1} - x_{i+1} y_i\\right)}.$$\n\n## Remarks\n\nWithout the absolute value, the sum gives the signed area, which is positive when the vertices run counterclockwise.\nEach 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 $P$.\nEquivalently, the formula is [[Green's theorem|0HR3N02317]] applied to the line integral $\\frac{1}{2}\\oint (x \\, dy - y \\, dx)$ along the boundary.\nExample: for the triangle $(0, 0), (4, 0), (0, 3)$ the sum is $0 + 12 + 0 = 12$, so $A = 6$.\nSimplicity is needed: for a self-crossing figure eight the two lobes have opposite orientation, and the sum gives the difference of their areas.\n\n## Explanations\n\n### L1 human\n\nThe shoelace formula computes the area of a polygon from the coordinates of its corners alone.\nWrite the coordinates in two columns, multiply them crosswise like the laces of a shoe, and take half the difference of the two totals.\n\n### L2 human\n\nThe formula is the standard way to compute polygon areas in computational geometry and surveying.\nIt is a discrete case of Green's theorem, and it is also the main tool in many proofs of [[Pick's theorem|K9SBQZMWB9]].\n",
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