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 ?
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.