We start with the given circle, center O.
Note: If you are not given the center, you can find it using the method shown in
Finding the center of a circle with compass and straightedge. 

1. Mark a point anywhere on the circle. This will be the first
vertex of the hexagon. 

2. Set the compasses on this point and set the width of the compasses to the center of the circle.
The compasses are now set to the
radius of the circle 

3. Make an arc across the circle. This will be the next vertex of the hexagon.
(It turns out that the side length of a hexagon is equal to its circumradius  the distance from the center to a vertex).


4. Move the compasses on to the next vertex and draw another arc. This is the third vertex of the hexagon. 

5. Continue in this way until you have all six vertices. 

6. Draw a line between each successive pairs of vertices, for a total of six lines. 

6. Done. These lines form a regular hexagon inscribed in the given circle. 
