- A student asks,
- How can I write the equation of a regular polygon using polar coordinates?

- MathHelp replies,

Consider the equation of just one side of an *n*-gon with radius 1. Better yet, consider the equation of just one half of one side of the *n*-gon. It's the side of a right triangle opposite angle *A* = π/*n*. The hypotenuse of this triangle is 1 (the radius of the *n*-gon), so the equation of this one side is cos(π/*n*)/cos(θ), where θ goes from 0 to π/*n*. The other half of this side has the same equation, so we're already at the point where we know the equation of one side of the *n*-gon:

*r*= cos(π/*n*) / cos(θ), -π/*n*≤ θ ≤ π/*n*

Next, we will need to develop a function of θ that "normalizes" θ to be within plus or minus π/*n*. This is accomplished by the "floor" function, as follows:

*B*= θ - 2 π/*n*floor((*n*θ + π)/(2 π))

Putting it together, the equation of an *n*-gon is *r* = cos(*A*)/cos(*B*), which is

*r*= cos(π/*n*)/cos(θ - 2 π/*n*floor((*n*θ + π)/(2 π)))