The circle of fifths has many applications. First off, I have my students write it out: 1. draw a circle of a piece of paper 2. beginning at 12 o'clock, write C, then progressing clockwise, you count up a 5th from 1 o'clock through 6 o'clock: C – G – D – A – E – B – F#. The right hemisphere of the circle deals in sharps and the left in flats.
To complete the left hemisphere, begin at 12 o' clock ( C ) and now proceed counter-clockwise in 4ths ( 4ths and 5ths are reciprocally inverse ) this produces: F – Bflat – Eflat – Aflat – Eflat – Dflat – Gflat.
I suggest students write this out everyday for a week or two by then you'll have a good handle on it.
A rudimentary use for the circle of fifths is to identify the accidentals in keys, here's how: G is at 1 o'clock and has one sharp, D is at 2 o'clock and has two sharps and so on ( A = 3 sharps, E = 4 sharps, B =5 sharps, F# = 6 sharps ) In other words, the numbers of sharps on the right hemisphere of the circle corresponds to the position on the numbers on the clock that are replaced by letters.
The left hemisphere deals in flats, to find them, proceed counter-clockwise from C ( 12 o'clock ) progressively adding one flat with each successive step on the clock. Ex. 11 o'clock = one flat, 10 o'clock = 2 flats, 9 o'clock = 3 flats and so on.