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

IA idea · Sport

Where should a winger shoot from? Maximising the angle to goal

AA SLAA HLAI SLAI HL Solid Also in: Calculus, Geometry & Voronoi

Research question

Running parallel to the touchline at a given distance from the goal line's centre, at what point is the angle subtended by the goal largest, and does shot data support shooting from there?

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Why it makes a good exploration

The goal looks biggest from a particular spot on a winger's run — the same mathematics as the best cinema seat and the rugby conversion problem. Linking the geometry with real shooting data makes it a sports-analytics IA.

The mathematics you'll need

Inverse trigonometric functions for the subtended angle
Differentiation to find the maximum
Showing the optimum is where a circle through the posts touches the line
Comparing with shot data

Where the data comes from

Use pitch dimensions from the laws of the game; film or chart shots in your own matches.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

A possible outline

Set up coordinates on the pitch.
Write the angle as a function of distance run.
Maximise it.
Prove the tangent-circle property.
Compare with where goals are scored in your matches.

Pitfalls that cost marks

Treating a larger angle as the only factor in scoring.
Wrong pitch dimensions.
No link to real play.

Showing personal engagement

Chart your own team's shots.
Compare with rugby conversion kicks.
Advise a winger in your team.

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	Inverse trigonometric functions for the subtended angle; Differentiation to find the maximum
AA HL	Good fit	Inverse trigonometric functions for the subtended angle; Differentiation to find the maximum
AI SL	Good fit	Inverse trigonometric functions for the subtended angle; Differentiation to find the maximum
AI HL	Good fit	Inverse trigonometric functions for the subtended angle; Differentiation to find the maximum

Level: Solid. Needs some independent work beyond class examples. 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: treating a larger angle as the only factor in scoring; and wrong pitch dimensions. Say how each one affects your answer.

Use of mathematics (E)

SL: inverse trigonometric functions for the subtended angle; differentiation to find the maximum; showing the optimum is where a circle through the posts touches the line; comparing with shot data, 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

Add the goalkeeper's position and find the angle of the open goal instead.

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, AA HL, AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Inverse trigonometric functions for the subtended angle, differentiation to find the maximum, showing the optimum is where a circle through the posts touches the line and comparing with shot data.

Where can I get data for this IA?

Use pitch dimensions from the laws of the game; film or chart shots in your own matches. Checked sources and tools: GeoGebra.

Turn this idea into your IA

The planner opens with “Where should a winger shoot from? Maximising the angle to goal” filled in: the question, the maths and the data sources.

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