A diagram of the cycle of 5ths is useful as a teaching aid - particularly if you can show someone that a lot of basic jazz chord progressions (for example) are based on the principle of moving from the tonic chord to a point on the circle, and then working back round the circle to the tonic chord. "Sweet Georgia Brown", "Ain't She Sweet" and a thousand others of their ilk fit this style.

However, once the learning phase is over, the important thing is to understand those chord relationships on the instrument - particularly if you're playing jazz.

Example: One of the tunes in the rep of the jazz outfit I played in, many years ago now, was "All The Things You Are" (Jerome Kern) which - if you know it - you'll remember drags you kicking and screaming through the circle of 5th in no uncertain terms! The 'standard' key for that tune is Ab - and that's what we played it in. One evening, a visiting tenor player (the late Danny Moss) sat in with us and chose that tune to play - in Eb! As custom dictated, the guest had the choice of tune and key.

Now, you could attempt the transposition - on the fly - by visualising the cycle of 5ths... But, actually, it's much better to know the relationship of chords in a progression on your instrument - in my case, guitar. I always visualise chord progressions as a series of geometric paths across the fretboard across, up and down the fretboard. Instant transposition then becomes a different starting point on the geometric path. Sounds complex, but experience makes it OK after time.

If you're interested, here's the usual (simplified) chords for "All The Things You Are" in Ab:

Fm / / / Bbm / / / Eb / / / Ab / / /
Db / / / G7 / / / C   / / / C   / / /
Cm / / / Fm / / / Bb7 / / / Eb / / /
Ab / / / D7 / / / G   / / / G   / / /
Am / / / D7 / / / G   / / / G   / / /
Am / / / B7 / / / E   / / / E+ / / /
Fm / / / Bbm / / / Eb / / / Ab+ / / /
Db / / / Dbm / / / Ab / / / Abdim / / /
Db / / / Eb9 / / / Ab / / / Bbm / C7 /

As you can see, you get taken through the chords in a beautiful way. Now - without opening your book - transpose and play instantly on the guitar... in Eb.