The center of gravity, also called the center of mass, of a rigid object can be found by calculating the point at which the torques applied by the mass of the extended object sum to zero. If you support the object at that point, gravity will not cause it to rotate. And this will happen regardless of the orientation.
A simple example is the wheel. If you support it by only the axle, it will not spin.
There is a nice trick with an arbitrary flat board. Take a sheet of cardboard and support it from a single hole somewhere around the perimeter. Draw a line from that point directly down. Next hold the cardboard from another hole. Draw a line directly down from that point. You can repeat this as many times as you like. All of the lines will intersect at a single point. This is the center of gravity.
If the object has some concave side or a hole, the center of gravity may not be inside the object. This is the case with a hoop and many crescent shapes.