Why can't we write an equation for a polygon?

Question

You can write an equation for a circle, but why can't you write an equation for a triangle or any other polygon? By equation I mean an equation that is not just a piecewise equation of lines.

Answer 1

Polygons are made of line segments. A parametrization would encounter a discontinuous derivative moving from one segment to the next. Inherent in the construction of polygons is that they are pieces of lines, segmented and attached to one another. So you can expect a piece-wise description.

Circles neither have these discontinuous nor are they even composed of line segments.

Answer 2

They can be defined by equations. The only issue is what kind of functions you are willing to use in the equations. We can use artificial piecewise-defined functions. More "natural" would be the so-called Schwarz-Christoffel mappings in complex analysis, with some finagling I imagine.

Perhaps you want to restrict your attention to algebraic equations. If any line segment is in an algebraic set (the zero loci of some algebraic equations), then the whole line is as well. One way to see this is to write the equation(s) as $f(x)=0$, then parametrize the line and plug into $f(x)$ and finally differentiate with respect to the parameter: since the derivative is a polynomial that vanishes on an interval it must be everywhere zero. As polygons have line segments which are not complete lines, they cannot be the solution set to a system of algebraic equations.