The chords of a key can be worked out from its scale, one triad at a time. They can also just be read off a wheel, and the wheel is worth having because it shows you something the list does not: why those particular chords belong together.
Turn it
Every key on the outside is a fifth from the one before it, and every relative minor sits directly inside its major. Tap anything to move the wheel there.
- IV4F
- ii2-Dm
- I1C
- vi6-Am
- V5G
- iii3-Em
The same six, under your hands
Turning the wheel changes these too. Three majors and the three minors that live inside them, all near the nut, which is as much of the guitar as a great many songs ever ask for.
IVF iiDm IC viAm VG iiiEm
Every chord is held by all five CAGED forms, spread up the neck, so each of these has the same chord waiting nearer the nut and further up. Some do not: an open E has nothing above it, because there is nothing above the nut. The loop starts again at the twelfth fret, where the form you began with is waiting.
Three slices, not six chords
The six chords of a key are never scattered around the wheel. They are always three touching slices: the key itself, its neighbour on each side, and the three minors underneath them. That is the whole picture, and it is the same picture in every key.
So F, C and G go together not because somebody decided they sound nice, but because they are next to each other on a wheel of fifths, and neighbouring keys share all but one note. Move one notch clockwise and every chord moves with you: the shape of the family never changes, only its position.
What it is for
Working a song out, the wheel tells you where to look first: the chord you cannot name is nearly always one of the five sitting next to the one you are on. Writing one, it tells you what is available before you go looking outside the key. And transposing, it is the whole job: find the new key, and every chord keeps its place.
The seventh chord of a key, the diminished one, has no slice. It is built on the note that does not belong to any neighbouring key, which is another way of saying the same thing the 7° always says: it is the odd one out.