There are a few ways to find the center of a circle when you have the circle already (e.g., a round table top that's already cut, a circular hole, etc.)

 

Method 1: Framing square and straightedge
Lay a framing square down so the outside point is on the edge of the circle.  Mark the points where the two legs cross the  perimeter (use the outside of the legs).   Those two points will define a diameter, so connect them with a straightedge.   Repeat for another point.  Where the two diameters cross will be the center.  In case you have some "roundoff error" it doesn't hurt to do a third point and third diameter.  If you are off a little, there will be a small triangle in the middle.  The center of the triangle should be the center of the circle.

Method 2: Straightedge and framing square.
With the straightedge, mark a chord across the circle.  For ease, make it a whole, even number like 24" not 23 13/16"  Mark the center point of the chord.  Lay the square with one leg on the chord and the point at the midpoint of the chord.   That second line will go through the center.  Repeat once or twice more as for method 1 to find the center.

 

Method 3: Compass and straightedge

This will work for larger and smaller circles well, but requires having a point you can use both inside and outside the circle.   It can just be a piece of scrap set in the right position(s) for the below. Set the compass to something less than a estimate of the radius.  For example, if you have a 10" radius circle, set your compass to 7 or 8", it really does not need to be anything accurate.  Set the compass point on the perimeter of the circle and mark an arc both inside and outside the circle.   Move to another point on the perimeter that's a bit less than the compass distance from the first spot.   Make two arcs like before that intersect the first two arcs.   Connect those points with a straightedge.  That will be a diameter.   Repeat one or two more times and use the diameters to confirm the center point.

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