geometrypolygons
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?
For a $n$ sided polygon, you need all the angles in order and $n-2$ consecutive side lengths in order to construct the polygon. So, you need the lengths of sides $B,C$ or $E,B$ or $D,E$ to construct your polygon.
The best way to find out the length of the remaining side is by drawing diagonals and applying triangle laws (sine or cosine rule).
Consider the (very badly drawn) pentagon. It is not drawn to scale, but you get the idea. Here are the steps you will take to find out the lengths of $D,E$.
1.Find out length of $X$ using cosine rule in $\Delta ABX$.
2.Knowing $\angle a,X,A,B$, find out $\angle e,\angle b$ using sine rule.
3.$\angle c = \angle\mbox{(between B,C)} -\angle b$. So, $\angle c$ is known.
4.Repeat the whole procedure for $\Delta CXY$. Find out $Y,\angle d, \angle f$.
5.$\angle g, \angle h$ are easily calculated now.
6.$\angle i$ is known. Apply sine rule in $\Delta DEY$ to find out $D,E$-the two unknown sides.
You have too many degrees of freedom. The length of the sides and the area are enough to determine the configuration for a pentagon but not for an octagon.
Best Answer
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.