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Is there a formula for the area of a general polytope with $n$ vertices which just uses the distances between vertices, like Herons formel for the area of a triangle?

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    $\begingroup$ Regular polygons or any polygon? $\endgroup$ – Stefan4024 Sep 2 '13 at 19:15
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In general, the area of an $n$-gon cannot be deduced from the distances between vertices alone. For example, the two $8$-gons

n-gons with different areas

have the vertices in the same places (and thus the distances between vertices will be the same), but will have different areas.

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Your question mentions polytopes, so you could be asking whether Heron's formula generalizes to higher $n$-dimensional simplices. For such a simplex, every pair of points is connected by an edge, and unlike the context of Rebecca's answer, the resulting shape is uniquely determined by all of these distances. There is indeed a more general formula, given quite elegantly by Cayley-Menger determinants.

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