Natural angle: the 30° and 90° rulesBall first — the direction of the cue ball after ball 2

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ScalesReference linePath

Reference positionBall fraction 1/2Cue ball at contact Natural roll (30° rule)Deflection angle (°) 33.7

Spinside 0 · high

1/2

Ball fraction1/2

SpeedNot specified

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Half ball, natural roll → the cue ball deflects 33.7° from the aiming line. Ball 2 goes 30° to the other side.

Which way does the cue ball go after it hits ball 2? This is decided by the ball fraction and by the vertical spin at the moment of contact. A cue ball that has no forward or backward rotation at contact (sliding) leaves perpendicular to the direction in which ball 2 travels: the 90° rule. A cue ball that strikes with natural roll, on the other hand, deflects about 30° from the cue line at fractions between 1/4 and 3/4 ball: the 30° rule. The largest deflection, about 34°, occurs at half ball; that is why half ball is known as the 'natural angle'.

When? On ball-first shots, to estimate the first direction of the cue ball after ball 2, i.e. where it will go to the 1st cushion. Especially in positions played half ball that you want to be forgiving of small aiming errors.

✓ 6/6 examples verified by the formula

Calculate

Sliding ball: Deflection = 90° − cut angle · Rolling ball: tan(Deflection) = sin θ · cos θ ÷ (sin² θ + 0.4)

Ball fraction

Cue ball at contact

Deflection angle (°)33.7

Deflection angle (table value)33.7
Deflection angle (°)33.7

✓ Example: Reference position: half ball, rolling

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Watch outThe values are for a straight stroke without side spin. Side spin also deflects ball 2 slightly and changes the path of the cue ball after the cushion.

Watch outWith a hard stroke the cue ball can reach ball 2 before it picks up natural roll. The 30° rule holds when the ball is rolling naturally at contact: a soft stroke, a hit above center, or enough distance makes sure of this.

Watch outThe values in the table are theoretical; on a real table ball-to-ball friction and cloth conditions can change the result by a few degrees.