full ball -- 0 degrees -- straight shot

3/4 full -- 15 degrees

1/2 full -- 30 degrees

1/4 full -- 45 degrees

1/8 full -- 60 degrees

Except for the half-ball hit which is 1/2 full and a cut of 30 degrees, those numbers are all slightly off, but the errors are not important for this first part of the discussion. Another factor that must eventually be considered is throw which varies with the speed and spin of the cue ball but that gets very complicated and it is better to completely understand the simple fundamental ideas of the situation before we get into all the nasty details. Those details are important and we'll get to them.

In the diagram you see a shot off the spot and five different areas where the five fractional aims will be useful. Let's consider the straight shot or 0 degree cut. How wide is that area? The pocket is two balls wide (more or less) and the two extreme arrival positions of the object ball are shown by the 2 ball and the 3 ball. If you draw straight lines from the centers of those balls back to the 1 ball, you get the angular width of the pocket. It is about 4 degrees. If the object ball is sent more than 2 degrees away from dead center of the pocket, it won't go in. (Again, this is not perfectly accurate. From this approach angle, the pocket is larger for very hard shots, so the pocket size varies with the shot conditions. Pick your own number if you don't like 4 degrees of pocket width.)

Where can the cue ball be in order to pocket a straight shot to that pocket off the spot? It is pretty obviously the red shaded area marked "0°". If the cue ball is anywhere on the bottom/right side of the shaded area, with a perfect, full hit the object ball will pass over the position of the 2 ball. If the cue ball starts on the top/left side of the shaded area the cue ball will pass over the position of the 3 ball. If the cue ball is anywhere outside the red shaded area, a full hit won't pocket the object ball.

The same argument applies to the other standard fractional cuts. It is not hard to see that all the triangular areas for the different cuts are 4 degrees wide. This means that the standard fractional cuts for a shot as hard as a spot shot cover only about 1/4 of the area of the table. The "good" triangles are 4 degrees wide and 15 degrees apart.

But it is a really, really bad idea to wed yourself to exact fractional ball hits. Just consider the straight shot. If the cue ball is on one edge of the red shaded area, and you use a true fractional aim, you will send the 1 ball all the way to the extreme side of the pocket. If your stroke makes a small error in that same direction, you will miss the shot. Since most of us make small errors most of the time (and large errors the rest of the time ), you will end up missing about half of such shots.

The lesson from this is that the "good" triangular areas for the shot are actually considerably smaller. If you are willing to give up half of the allowed error to your aiming system, then the triangles will shrink to 2 degrees wide and the percent of the table that is covered by these fractional aims drops by a factor of two to roughly 1/8th or 12.5% of the table surface.