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. Draw a diameter of the circle through the center point and mark its endpoints C and M.
It does not have to be vertical. 

2. Construct a perpendicular to CM at the point O.
For more on this see Constructing a perpendicular at a point on a line.


3.
Mark the point S where it crosses the circle.


4.
Find the midpoint L of the segment SO by constructing its perpendicular bisector.
For more on this see Constructing the perpendicular bisector of a line segment.


5. Set the compasses on L, adjust its width to S or O, and draw a circle. 

6.
Draw a line from M, through L so it crosses the small circle in two places. Label them N and P. 

7.
Set the compasses on M and adjust its width to P. 

8.
Draw a broad arc that crosses the given circle in two places. Label them A and E. 

9.
Set the compasses on M and adjust its width to N. 

10.
Draw a broad arc that crosses the given circle in two places. Label them B and D. 

11.
Draw a line from A to B, then B to C etc, until you have drawn all five sides of the pentagon. 

Done.
ABCDE is a regular pentagon inscribed in the given circle.

