I studied music theory for a good long time before it dawned on me that you can read the major scale right off the circle of fifths. Here’s the C major scale on the circle. The white notes are the ones in the scale and the black ones are the ones outside the scale. The white notes correspond to the white keys on the piano, and the black notes to the black keys.
The circle of fifths trick works for all the major scales, not just C. Pick any note on the circle and think of it as the root. The note immediately counterclockwise from the root will be the fourth of the key. The five notes going clockwise from the root are the fifth, second, sixth, third and seventh of the key respectively. The other five notes will be the ones you omit — the “black keys.”
Another intriguing thing about seeing the scale written out on the circle of fifths is that it makes it easy to derive all the parallel major scale modes. Take a look at the two notes on the ends of the arc of white keys: F and B. If you bring the B down a half step, you get C Mixolydian. If you bring B and E down a half step, you get C Dorian. Bring B, E and A down a half step and you get C natural minor. Bring all of the above plus D down a half step gives C Phrygian, and bringing all of those plus G down a half step gives you C Locrian.
The circle of fifths suggests some nice chords, too. If you play the white notes at once from F to B, going clockwise, you get a beautiful voicing for Fmaj7(#11) with all of its extensions. If you leave off the F and go clockwise from C, you get a nice Cmaj7, again with all the extensions. If you go clockwise from G, you get G6/9. Go clockwise from D, and you get an ambiguous D6/9 or D-13.
If you go counterclockwise, each group of four notes gets you the fourths chord beloved of modern jazz. Starting on B gets you B-11, starting on E gives E-11, and so on.
Seeing the major scale on the circle of fifths gives a window into George Russell’s Lydian chromatic concept of tonal organization. Russell saw the Lydian mode as the “center of tonal gravity” for the key of C. Bringing in non-diatonic note from further around the circle gives a feeling of moving progressively further away from the center of gravity. Use your ears and experiment.