How do alchemists draw transmutation circles? In FMA, alchemists draw transmutation circles that are made up of perfect circles and regular polygons. 



Is it ever show how they make perfect circles and entirely equal angles? I know that it is difficult to free-hand draw things like that. Is anybody ever shown using rulers/compasses, or is it just understood that they are all able to free-hand circles/polygons without them?
First, there is no official word on this. None of the Hiromu Arakawa (the original writer) interviews go into depth as to what the transmutation circles even do, much less how they are created. Neither of the FMA Perfect Guide books mention transmutation circles to any strenuous degree, though it is mentioned that Alphonse has more practice drawing them than Edward (for reasons obvious to those who have seen the series).

There are tutorials online that show methods to draw these circles; however, most if not all that I found used rulers, compasses, or other tools.

Some alchemists use circles that are likely made by tools:


Major Armstrong's circles are inscribed in the metal plates on his hands. These would require tools to make, and so may have been made perfectly circular by tools as well.
The same applies to other circles we see; Roy's are on his gloves, Basque Grand's are on his gauntlets, etc.


However, alchemists are shown drawing the circles many times:


In a flashback during episode 2FMA:B (and others), Ed is shown to draw a transmutation circle (the blood seal) on Alphonse's armor just using his finger, and he's still a child.
Around episode 23FMA:B, May Chang is shown to draw a "perfect" circle and pentagram using only her foot. She then proceeds to throw ten kunai into two perfect pentagons, which are used as transmutation circles.
In episode 57FMA:B, the Gold-Toothed Doctor uses chalk to draw a rather large transmutation circle.
In episode 63FMA:B, Edward Elric draws a large transmutation circle by only dragging a pipe across the ground.
In chapter 1Manga, Al is seen drawing a circle using chalk and no tools (pictured below). This shot appears in similar fashion in Brotherhood.




In none of the above circumstances are tools shown to be used to draw the circles. Furthermore, Scar's brother's tattoo likely could not have been drawn with tools as the human skin is not a uniform or Euclidean surface. And also, the Elric brothers are shown, prior to their mother's death, to draw effective transmutation circles without tools.

With all these bits of information together, it's at the very least hard to believe that there are tools used to draw the circles, or even to learn to draw them. No one is ever seen using them; I suspect we are just supposed to believe that freehanding it becomes a skill as alchemists get better.