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Let the length of sides and internal angles be the $2n$ paramaters defining a n-gon. How many of them are needed to define it completely without over specifying?

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In "A treatise on Algebra" George Peacock stated that a polygon is defined by three equations. Let $a_i$ (with $i=1,2,..,n$) be the lengths of the sides; and $\theta_{i,i+1}$ the internal angles between them. Then:

$$a_1+\sum_{i=2}^{n}(-1)^{i-1}a_i\cos\left[\sum_{j=2}^i\theta_{j-1,j}\right]=0\\ \sum_{i=2}^{n}(-1)^ia_i\sin\left[\sum_{j=2}^i\theta_{j-1,j}\right]=0$$

are known as the "equations of figure". While $$\sum_{i=1}^n\theta_{i,i+1}=(n-2)\pi$$ is the "equation of angles".

These equations make up for not knowing up to three angles. Therefore a polygon is completely determined by $2n-3$ parameters. However they can only be split in three situations: $n$ sides and $n-3$ angles, $n-1$ sides and $n-2$ angles, and $n-2$ sides and $n-1$ angles.

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  • $\begingroup$ So the degrees of freedom of an n-gon is $2n - 3$. $\endgroup$ – Angelorf Oct 16 '18 at 8:48

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