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Are there any Heron-like formulas for convex polygons ? By Heron-like I mean formulas without angles as arguments and which takes as arguments only lenghts of sides of polygon - that is - we know no lengths of diagonals ? Does such formulas exist ? I don't think so, because we could go with area of regular quadrangle to zero, but how to prove it for convex k-polygon, or mayby in other cases it isn't true ?

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3 Answers 3

There are such formulas for cyclic polygons (i.e., those inscribed in a circle). I draw your attention to the work of Robbins in this respect.

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No. Consider a quadrilateral with all sides equal. You can vary the angles and change the area without changing the side lengths. Consider a rhombus and a square with equal side lengths, for example.

Similar examples work in the case of a general regular $n$-gon.

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I've wrote that. –  Qbik Mar 6 '13 at 2:21
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@Qbik What exactly do you want then? No such formula exists. –  Potato Mar 6 '13 at 2:49

Your intuition that such a formula cannot exist is correct. For general quadrilaterals or other polygons you may not be able to get the area to zero, but all you need is that it can vary depending on one angle.

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