Mark three points anywhere on the circumference of the circle. Designate these points A, B and C. It doesn't matter where you put these points, but the farther they are from each other, the more accurate your calculations will be
Draw a line connecting points A and B. Call this line AB. Do the same with points B and C to create line BC.
Find the center of line AB and BC. You can either measure the distance and divide it in half or use geometry and bisect the lines. You can use a compass set at a distance greater than half the length of the line segment and make arcs above and below the line segment. Do this from both endpoints to form points above and below the line segment, then connect these points to divide the line segment in half. Mark these middle points D and E.
Draw perpendicular lines from points D and E using the compass technique or using a protractor to find a 90-degree angle. Where these lines intersect is the center of the circle.
You can check that you have found the center of the circle. Simply measure from the point you found to the edge of the circle. Pick a few different points. If they are all the same distance, you have found the center.
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measure it
Draw a cord through the circle (a line through the circle, but not too close to where you imagine the center to be). With construction techniques, find the perpendicular at the center point of the cord, and draw the perpendicular. Do the same thing again starting with a different cord, and the two perpendiculars will intersect at the center of the circle.
You would have to know the length of the radius. The center of the circle is at one end of the radius. If you just know where some part of the radius is, and not that the part touches the circle then you cannot know where the center is without at lest a point on the circumference.
a diameter
You cannot. The circle could be located anywhere on a plane.