The two bar chord shapes you have studied so far, root 6 (E formation) and root 5 (A formation), involves sliding an open chord along the fretboard, i.e., E open and A open respectively. This concept could be applied to all open chords, but due to awkward fingerings only a few shapes are practical.
One of the most useful chords to slide up the guitar is C major. Hold an open C chord, using the 2nd, 3rd and 4th fingers (see first diagram).
Now slide this shape up one fret, and bar with the first finger (diagram two):
C
Db
This chord shape has its root note on the 5th and 2nd strings, and shall be referred to as a "C formation" bar chord. As with the other bar chords, it can be moved along the fretboard, changing name with each new position.
Try the following exercise, using the C formation bar chord in conjunction with root 6 and root 5 bar chords.
Note that the C formation bar chord is extremely useful in a major to relative minor chord progression.