You can easily find the exact center with just a ruler.
Pick a length of a chord that makes sense for the size of the circle that you are dealing with that makes an even measurement, say 1" for example.
Move the ruler along the radius till two points of the circle match your dimension (1" here). Draw two chords that are 1" making an extra tick at the half way mark, 1/2" in this example.
If your ruler has a square bottom, make a perpendicular line from your half way mark on each line toward the center of the circle. The two perp lines will cross, and X marks the spot.
Heres a quick sketch of this method in Autocad, on a 2" diameter circle using the example numbers above.