I can plot a thin spiral using just:

$ r=\frac{\theta }{2 \pi };$

 PolarPlot[theta/2 Pi,{theta,0,2 Pi}]

but is there a way to plot a spiral that has actual area (not just a visually thicker line, but actual 2D spiral as if you took a long rectangle and curled it up, if that makes sense...)? Like these:
![enter image description here][1]

I will need to convert the set of points that makes up such a spiral into a 2D array later, but I need to know how to plot it first.
I guess maybe it's not so much of a plotting question, as much as is there an equation that would provide such a result?

 [1]: https://i.sstatic.net/KMo89.jpg