Does specifying the lengths of the sides of a polygon completely fix its shape (area and angles)? Mine is a very generic question. I believe the answer is yes for a triangle. I don't have a formal proof, just an image in my head.
Is it also true for a higher $n$-gon? Does anyone know of a theorem?
EDIT (following comments):
The answer is NO for $n>3$. This is because the lengths of the sides of an $n$-gon do no specify its angles, for $n>3$. E.g., consider the square and the rhombus having the same sides. 
FOLLOW-UP question: 
Why is it this so? What is special about $n=3$, that changes for higher $n$. 
General guess to an answer:
I am a physicist, so let me use a physicist's argument. The shape of an $n$-gon is specified by its $n$ sides and $n$ angles. That's $2n$ parameters. Somehow, for $n=3$, there are sufficient constraints such that all 3 angles are determined by the 3 sides. For higher $n$, that is not the case, so that we are left with extra parameters, and can get a range of shapes. But why is this so? Someone please fill in the blanks.
 A: The question is one of the first asked about the rigidity of bar and joint  frameworks. For polygons in the plane the degree of freedom count in the comments settles the question (almost):
$n$ points in the plane have $2n$ independent degrees of freedom. If you subtract the three degrees of freedom corresponding to the Euclidean motions (translation and reflection) that leaves $2n-3$. So most of the time the configuration will be rigid if you add that many bars between pairs of points in some reasonable way. Adding diagonals to triangulate a convex polygon is one of those reasonable ways. The $n$ edges and $n-3$ diagonals do the job. 
But there are both mechanical and combinatorial problems you have to avoid. For example, you can add $3$ diagonals to a hexagon but if they involve just $4$ of the vertices you'll still have a motion.
The mathematics to begin to formalize these notions and prove theorems is very elementary physics and a little linear algebra. Since I'm not really "filling in the blanks" I'll mark this answer community wiki.
You can start reading by searching the web for "bar and joint frameworks". Here are some links:
https://en.wikipedia.org/wiki/Structural_rigidity
http://www.math.cornell.edu/~connelly/Connelly-Jordan-Whiteley.pdf
http://maths.nuigalway.ie/~jc/docs/NUIG2.pdf
