Pool Cut Angle & Ghost Ball Calculator
Ball fraction and ghost-ball offset for any cut angle, how much aiming error the shot allows, and where the 30-degree rule stops being true.
The cut angle is between the cue ball's path and the object ball's path to the pocket — zero is straight in. Defaults are American pool: 2.25″ balls and a 4.5″ corner pocket. Snooker is 2⅛″ balls; a bar box often has a tighter mouth than a nine-foot table.
All of this is the geometry of an ideal collision. Cloth throws the object ball a degree or two away from the contact line — most of it near a half-ball hit, and more with a dirty ball or slow speed — so treat the aim as the starting point your eye corrects from, not a number to chase to three decimals.
Fractions of a ball, and the angles they cut
Sighting down the cue ball's path, the two balls cover overlapping bands of width 2R whose centres are 2R·sin θ apart. The overlap is 2R(1 − sin θ), and as a share of the object ball that is just 1 − sin θ. Every fractional-aiming reference in every instructional book is that one line evaluated:
| Hit | Cut angle | Ghost ball offset | Error multiplier |
|---|---|---|---|
| full | 0° | 0.000″ | ×1.00 |
| 7/8 | 7.18° | 0.281″ | ×1.01 |
| 6/8 | 14.48° | 0.563″ | ×1.03 |
| 5/8 | 22.02° | 0.844″ | ×1.08 |
| 4/8 | 30° | 1.125″ | ×1.15 |
| 3/8 | 38.68° | 1.406″ | ×1.28 |
| 2/8 | 48.59° | 1.688″ | ×1.51 |
| 1/8 | 61.04° | 1.969″ | ×2.07 |
⭐Which explains something players learn by feel and rarely see stated. A half-ball hit is the reference not because 30° is a round number but because 2R·sin 30° = R exactly — the ghost ball's offset is precisely one radius, so the cue ball's centre is aimed dead at the object ball's edge. It is the only fraction on this table whose aim line lands on something you can actually see. Every other fraction asks you to aim at a point in empty space, which is why they are taught relative to this one.
Why thin cuts are hard, exactly
"Thin cuts are harder" is true and useless. The useful version is that the penalty has a closed form. The ghost ball sits at offset 2R·sin θ; misplace it by δ and the cut angle moves by δ/(2R·cos θ). So a given misjudgement of where the ghost ball goes is amplified into object-ball direction by exactly 1/cos θ — checked here by perturbing the offset numerically at every degree from 1 to 85 and comparing against the derivative, rather than trusting the algebra.
| Cut angle | Error multiplier | What it means |
|---|---|---|
| 0° | ×1.00 | straight in — errors pass through unchanged |
| 15° | ×1.04 | essentially free |
| 30° | ×1.15 | essentially free |
| 45° | ×1.41 | noticeable but forgiving |
| 60° | ×2.00 | every unit of sloppiness costs several |
| 70° | ×2.92 | every unit of sloppiness costs several |
| 75° | ×3.86 | every unit of sloppiness costs several |
| 80° | ×5.76 | the shot is mostly a precision test |
| 85° | ×11.47 | the shot is mostly a precision test |
Put real distances on it and the numbers get uncomfortable. With the cue ball and the pocket both 30″ away and a 4.5″ pocket, a 15° cut allows ±0.156° of aiming error. The same shot at 75° allows ±0.042° — about a 24th of a degree, or 22 thousandths of an inch of ghost-ball placement. That is the shot, described honestly.
The 30° rule is real, and narrower than you were told
A rolling cue ball does not leave along the tangent line, because its topspin survives the collision and friction turns it back downstream. Just after contact it moves along the tangent at V·sin θ while still carrying spin worth V in the original direction; a sliding ball settles at (5v + 2vspin)/7, which gives a deflection from the original path of arctan[ 5 sin θ cos θ / (5 sin²θ + 2) ].
Everyone quotes that as "about 30°". It is worth sweeping past where the rule holds rather than stopping inside it, because the curve has two edges and the folklore has none:
| Cut angle | Cue ball deflection | Is the rule usable? |
|---|---|---|
| 0° | 0.0° | no — out by 30° |
| 5° | 12.0° | no — out by 18° |
| 10° | 21.7° | no — out by 8° |
| 15° | 28.2° | yes — within 3° |
| 20° | 31.9° | yes — within 3° |
| 25° | 33.5° | roughly — within 5° |
| 30° | 33.7° | roughly — within 5° |
| 35° | 32.8° | yes — within 3° |
| 40° | 31.2° | yes — within 3° |
| 45° | 29.1° | yes — within 3° |
| 50° | 26.5° | roughly — within 5° |
| 55° | 23.7° | no — out by 6° |
| 60° | 20.6° | no — out by 9° |
| 65° | 17.4° | no — out by 13° |
| 70° | 14.1° | no — out by 16° |
| 75° | 10.6° | no — out by 19° |
| 80° | 7.1° | no — out by 23° |
| 85° | 3.6° | no — out by 26° |
| 90° | 0.0° | no — out by 30° |
The rule is good to within 3° only for cut angles from 14° to 49°, and to within 5° from 13° to 52°. It peaks at 33.75° near a 28° cut — above 30, not at it — and it falls to zero at both ends, because a straight-in shot stops the cue ball dead and a paper-thin cut barely touches it. Outside that band "30°" is a number read off the wrong part of a curve, and the error at a 70° cut is larger than the rule itself is precise.
The stun case needs no band at all: kill the roll and the cue ball leaves at 90° to the object ball, whatever the cut angle, because that component of momentum is the one the collision cannot transfer. It is the only piece of cue-ball control on this page that is exact everywhere, which is a fair argument for learning the stun shot before the rolling one.
What this leaves out
Throw, mostly. Friction between the balls drags the object ball a degree or two off the contact line, against the direction of the cut, and it is worst at slow speed, with dirty balls, and around a half-ball hit — which is exactly where fractional aiming is most used. Side spin adds its own throw on top and can be used deliberately to cancel the cut-induced kind. There is also squirt, where the cue ball leaves the tip off-line when you strike it off centre, and swerve, where it curves back. None of those are in the arithmetic above, and all of them are why the geometry gets you to the right neighbourhood and practice gets you the rest. A pocket 1.125″ wider than the ball on each side is the margin absorbing all of it.
How to use
- Enter the cut angle between the cue ball path and the line to the pocket.
- Add the two distances so the tolerance is for your actual shot.
- Read the ball fraction and the ghost ball offset to aim with.
- Check the error multiplier before deciding whether a thinner cut is the right choice.
Frequently asked questions
What is the ghost ball method?
You picture a ball sitting where the cue ball must be at the moment of contact — touching the object ball, directly opposite the pocket — and aim the cue ball at that phantom. It works because two equal balls send the object ball along the line joining their centres at contact, so placing that line through the pocket is the whole of aiming.
How do I convert a cut angle into a ball fraction?
The fraction of the object ball you cover is exactly 1 minus the sine of the cut angle. Sighting along the cue ball path, the two balls cover bands whose centres are 2R sin θ apart, so the overlap is 2R(1 − sin θ), which as a share of the object ball is 1 − sin θ. A half-ball hit is 30 degrees, three-quarter ball is 14.5, quarter ball is 48.6.
Why is a half-ball hit the standard reference?
Because 2R sin 30° equals R exactly. The ghost ball offset on a half-ball hit is precisely one ball radius, so the cue ball centre aims dead at the object ball edge — the only fraction whose aim point is a real thing you can see rather than a spot in empty air. Every other fraction is taught relative to that one.
Why are thin cuts so much harder?
The penalty is exactly 1 over the cosine of the cut angle. Misplace the ghost ball by some amount and the object ball direction moves by that amount divided by 2R cos θ, so a 60-degree cut doubles your error, 75 degrees nearly quadruples it and 85 degrees multiplies it elevenfold. It is not a feeling — it is a derivative.
How precise does my aim need to be?
Less than you would like. With a 2.25 inch ball, a 4.5 inch pocket, and the cue ball and pocket each 30 inches away, a 15 degree cut allows about 0.14 degrees of aiming error and a 75 degree cut allows about 0.04 — roughly twenty thousandths of an inch of ghost ball placement. The margin shrinks with distance at both ends of the shot.
What is the 30 degree rule in pool?
That a rolling cue ball leaves at about 30 degrees from its original path after contact. It comes from the cue ball keeping its topspin through the collision: it starts along the tangent line and friction turns it back downstream, settling at arctan of 5 sin θ cos θ over 5 sin squared θ plus 2.
Does the 30 degree rule always work?
No, and the band is narrower than it is usually taught. It is good to within 3 degrees only for cut angles from about 14 to 49, peaking at 33.7 degrees near a 28 degree cut — above 30 rather than at it. It falls to zero at both ends, because a straight shot stops the cue ball dead and a paper-thin cut barely moves it. At a 70 degree cut the rule is out by more than its own precision.
Where does the cue ball go on a stun shot?
Along the tangent line, at exactly 90 degrees to the object ball, whatever the cut angle. That one is exact rather than approximate — it is the component of momentum the collision cannot transfer — which is a decent argument for learning stun before natural roll.
What is throw and why is it not in the calculation?
Friction between the two balls drags the object ball a degree or two off the contact line, against the direction of the cut. It is worst at slow speed, with dirty balls, and near a half-ball hit, which is exactly where fractional aiming gets used most. Modelling it well needs ball condition and speed, so this page gives you the ideal geometry and flags where reality departs from it.
Does this work for snooker and English pool?
Yes — change the ball diameter and pocket width. The geometry is scale-free, so the fractions and angles are identical; only the tolerances move. Snooker balls are 2 1/16 inches and the pockets are tighter relative to the ball, which is why the same shot is a harder shot.
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