I have only lengths for the sides of an irregular polygon, can anyone tell me how I can measure the area of the polygon? Remember only lengths of all the sides , no angles or coordinates.
Few forums mention about trangulation of the polygon etc But I only have side lengths.
Does anybody has any feedback?
On
The area of a square with all sides $1$ is greater than the area of a parallelogram with all sides $1$, if the parallelogram is not a square.
On
Building on coffeemath`s answer i think i should mention this (because he uses the square which is regular polygon and you want to work with irregular ones):
It is true that for every $\varepsilon>0$ there exists parallelogram with side lengths all equal to $1$ such that its area is less than $\varepsilon$ so at least in the case $n=4$ we need more information then just the side lengths.
On
There is work by David Robbins, et al., on the area of cyclic polygons given the lengths of the sides.
Imagine building the polygon by joining pieces of wood with hinges. Because the structure is in general not rigid, the answer to your question is indeterminate except if:
(a) One side is longer than the sum of all the other sides - in which case no polygon exists
(b) One side is equal to the sum of all the other sides, in which case the polygon is degenerate of area $0$
(c) Your polygon is a triangle - apply Heron's Formula
Here is a proof that the polygon has maximum area if all vertices lie on a circle. It also proves that for the maximum area, the order of the sides does not matter.