You can also determine what key a piece is in empirically with a Circle of 5ths.

Suppose you have 3 flats in the signature. By the rules of music theory, you know that the order of the flats from left to right have to be B E A. First, we go to the Circle and bisect it such that B E A are flatted within a half-circle. In this case, we would draw the line from D to Ab. There are only three flats in the key signature and those have to be it and those note designations are now contained within a half-circle going clockwise and are the only flatted notes in it. Now we obtain the key by rotating clockwise to the next note on the circle from Ab, which is Eb. So the key signature with three flats is Eb major (or C minor).

Likewise, the other half-circle also forms a scale whose key is obtained by the same clockwise rotation. The other half-circle contains three sharps—F# C# G# (Ab)—and the key signature on the chart with three sharps is A, which is also clockwise from D.

What about no sharps or flats? We would draw a line from F to B, and isolate the unflatted and unsharped notes within a half-circle. C is clockwise from F so the key is C major. The other half-circle is the flatted side and so we use only the flatted note designations and there are six within the half-circle starting at Cb (B). Going clockwise from Cb to the next note on the circle is Gb and so that is the name of key signature with six flats in the scale.

Wouldn't it be easy to just memorize the keys? Of course, it is. That's not the point. The point is, that this information is contained with the Circle of 5ths if you know how to read it.