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The Shoelace formula determines the area of a polygon given its coordinates (see https://en.wikipedia.org/wiki/Shoelace_formula).

I am not too familiar with geometry, and I am looking for a reference for the shoelace formula, for example a geometry text book. Is there a standard reference for this?

Thank you.

Mathieu Rundström
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  • The usual proof relies on triangulating a general polygon but this topic is not covered in geometry textbooks: they may cover the convex case only if at all. – lhf Jun 16 '21 at 17:12
  • @lhf That makes sense. In this case, I am really just looking for any expository handling of the shoelace formula. – Mathieu Rundström Jun 16 '21 at 19:48
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    See https://math.stackexchange.com/questions/585145/proving-the-shoelace-formula-with-elementary-calculus and https://math.stackexchange.com/a/1103318/589 – lhf Jun 17 '21 at 00:37
  • No reference but my proof world go like this: $\frac12(x_1y_2-x_2y_1)$ it's the oriented area of the triangle formed by the points $(x_1, y_1)$, $(x_2, y_2)$ and $(0, 0)$. So the absolute value of that number equals the area and the sign depends on the orientation of the triangle. By following the edges of the polygon with this formula, you will get triangles that cancel out outside the polygon (both signs occur equally often), but leave a net contribution of $\pm1$ inside the polygon, with the sign depending on which direction you're walking the perimeter. – MvG Jun 17 '21 at 11:56

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