Pocket geometry: why a fast ball misses a pocket it should make

A WPA corner pocket is 4.5 to 4.625 inches wide at the mouth, and a pool ball is 2.25 inches across. On paper that is two balls side by side, which sounds generous. Then a well-struck ball hits the jaw and spins back out, and the shot that "should have gone" is left over the hole. The mouth is not the target. The target is much smaller, and it shrinks in two ways.

The mouth is not the target

Measure the mouth where the two jaw tips (the points of the cushion facings) sit. For a ball to drop without touching either tip, its centre has to pass between them with a ball radius, 1.125 inches, to spare on each side. So the window for the centre is the mouth width minus one ball diameter.

For a 4.5-inch corner pocket hit dead straight down the pocket's centreline, that is 4.5 − 2.25 = 2.25 inches. The ball can be off the centreline by 1.125 inches either way and still go in cleanly. A 4.625-inch corner gives 2.375 inches, and a 5-inch side pocket gives 2.75.

That is the best case. Few real shots arrive straight down the centreline.

The first shrink: the angle

When the ball arrives at an angle to the pocket's centreline, the two jaw tips are seen from a slant and the gap between them looks narrower. Treat the tips as two points. At an angle θ from the centreline, the gap seen by the ball is the mouth width times cos θ. The window is that minus the ball diameter.

For a 4.5-inch corner pocket:

• 0 degrees: 4.5 × 1 − 2.25 = 2.25 inches.
• 15 degrees: 4.5 × 0.966 − 2.25 = 2.10 inches.
• 30 degrees: 4.5 × 0.866 − 2.25 = 1.65 inches.
• 45 degrees: 4.5 × 0.707 − 2.25 = 0.93 inches.
• 60 degrees: 4.5 × 0.5 − 2.25 = 0 inches.

The window falls slowly at first and then quickly. From 0 to 30 degrees it loses 0.6 inch; from 30 to 60 it loses the remaining 1.65. At 45 degrees it is 41% of the straight-on window. At 60 degrees, where cos θ = 0.5, the gap equals the ball's diameter and the clean window closes. A 4.625-inch mouth stretches that to about 60.9 degrees, and a 5-inch side pocket to about 63.3.

Why shots past that angle still sometimes go

The point model is cautious. A real jaw is not a point but an angled facing leading into the pocket. A ball that clips it at low speed can be steered along the jaw and drop, which is why a gentle angled shot sometimes goes when the sums say it cannot. It is also the "rattle" you hear when a ball bounces between both jaws and either falls or spits out. Dr Dave Alciatore's well-known testing of pockets found that the usable range of angles depends heavily on speed and jaw shape, so there is no single number for every table. Treat 60 degrees as a rough guide to where clean entry ends.

The second shrink: speed

Pace does not change the opening. It changes what happens when the ball touches a jaw. A slow ball that kisses a facing is often turned toward the pocket. A fast ball bounces off keeping most of its speed, so it tends to cross to the far jaw and out again.

The practical rule has no number attached: the harder the ball is hit, the closer you must stay to the clean window, because a half-correct line is no longer rescued by the jaws.

• Straight-in shots tolerate pace best, because the whole 2.25 inches is available.
• Steep corner shots are the ones that spit. Past about 45 degrees the window is under an inch, so a fast ball needs near-perfect aim.
• Side pockets have a wider mouth, and the same formula gives 1.29 inches at 45 degrees against 0.93 for the corner.

What this means for choosing a shot

1. Play position for an open angle. A shot left at 20 degrees to the centreline has 4.5 × cos 20° − 2.25 = 1.98 inches of window, against 0.93 at 45 degrees: more than double.
2. Slow down on steep shots. With under an inch to work in, a fast ball punishes a small error. The ghost-ball post covers the aim, and the throw post explains why the line you aim along is not quite the line the ball takes.
3. Do not trust a margin you never measured. A pocket that looks twice the ball's width has a 2.25-inch window at best.

Practising it in the app

The Play listing for 8balls says you can adjust aim, shot power and cue-ball spin, and switch between overhead and aiming views. It does not describe how pockets are modelled, so treat what follows as a way to test your own judgement rather than the app's physics. Pick a ball near a corner pocket at a steep angle, play it soft and then hard, and tally how many of ten go in each way. The overhead view helps you judge how far the ball sits off the pocket's centreline. For another use of angles, see the bank shots post.