Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

IA idea · Sport

Where should a pool ball hit the cushion? The geometry of bank shots

AA SLAI SL Accessible Also in: Geometry & Voronoi

Research question

Does the reflection method (mirror image of the pocket) predict where to hit the cushion for a bank shot, and how large is the error in my real shots due to spin and cushion compression?

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Why it makes a good exploration

Bank shots are pure geometry — angle in equals angle out — until spin and the cushion get involved. Testing the geometric prediction on a real table gives clean data and honest reflection.

The mathematics you'll need

Reflection in a line and coordinate geometry
Equations of lines and intersections
Angles and trigonometry
Mean error and spread

Where the data comes from

Film 30+ bank shots from above with a phone and measure contact points and angles with Tracker.

Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

A possible outline

Derive the mirror-image method.
Compute predicted contact points.
Measure real shots.
Compare and analyse errors.
Reflect on speed, spin and cushion geometry.

Pitfalls that cost marks

Camera not directly above the table.
Varying shot speed unintentionally.
Too few shots.

Showing personal engagement

Use a table you play on.
Test fast versus slow shots.
Extend to two-cushion shots.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Reflection in a line and coordinate geometry; Equations of lines and intersections
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Reflection in a line and coordinate geometry; Equations of lines and intersections
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: camera not directly above the table; and varying shot speed unintentionally. Say how each one affects your answer.

Use of mathematics (E)

SL: reflection in a line and coordinate geometry; equations of lines and intersections; angles and trigonometry; mean error and spread, each explained in your own words and carried out correctly.

How sport explorations reach the top bands in general: Sport IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Model a correction for cushion compression and see if predictions improve.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL sport exploration adds.

See a complete IA, marked

How long does a skydiver take to reach terminal velocity? (AA HL) is a complete sport exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL and AI SL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Reflection in a line and coordinate geometry, equations of lines and intersections, angles and trigonometry and mean error and spread.

Where can I get data for this IA?

Film 30+ bank shots from above with a phone and measure contact points and angles with Tracker. Checked sources and tools: Tracker video analysis and GeoGebra.

Turn this idea into your IA

The planner opens with “Where should a pool ball hit the cushion? The geometry of bank shots” filled in: the question, the maths and the data sources.

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