IA idea · Trigonometry, navigation & surveying

The hidden angles of a hip roof: 3D trigonometry

AA SLAI SLAA HL Solid Also in: Geometry & Voronoi, Art & music

Research question

For a real hip roof, what are the angles between the sloping faces, the hip rafters and the horizontal, and can they be found from measurements taken safely from the ground?

Adapt it: change the place, the data or the comparison until the question is yours.

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Why it makes a good exploration

3D trigonometry is in both SL courses, but problems usually come from a textbook. A real roof gives you measurements, a model to build and a check.

The mathematics you'll need

3D trigonometry: angles between lines and planes
Pythagoras in 3D
HL: vectors and the angle between planes
Measurement from the ground using elevation angles
Error analysis

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Measure the footprint of a building and angles of elevation from the ground; never climb. Photographs help.

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

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Measure the footprint and elevation angles.
Build a 3D model with coordinates.
Calculate the key angles and lengths.
Check with an independent measurement or a model.
Reflect on errors and real construction tolerances.

Pitfalls that cost marks

Unsafe measuring; stay on the ground.
Confusing angle with a line and angle with a plane.
No check on the model.

Showing personal engagement

Use your own house or school.
Build a card model to scale.
Ask a builder how they cut hip rafters.

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	3D trigonometry: angles between lines and planes; Pythagoras in 3D
AA HL	Good fit	3D trigonometry: angles between lines and planes; Pythagoras in 3D
AI SL	Good fit	3D trigonometry: angles between lines and planes; Pythagoras in 3D
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: 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

Personal engagement (C)

Take the measurements yourself, choose the place, and compare at least two methods. Explaining why you chose each position or angle is engagement an examiner can see.

Reflection (D)

Analyse measurement error: how much does a small error in an angle change the answer, which method is most robust, and would you trust the result for the decision you are making? For this idea, start with: unsafe measuring; stay on the ground — say how it affects your answer.

Use of mathematics (E)

SL: Right-angled trigonometry, sine and cosine rules, bearings and 3D trigonometry applied correctly to your own measurements, with an error analysis.

HL: Error propagation with derivatives, vectors in three dimensions, spherical geometry derived rather than quoted, or an optimisation of where to measure from.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Find the total roof area and the angle that minimises material for a given volume.

Extending it for HL

This idea already has HL mathematics in it: vectors and the angle between planes. Use derivatives to find how sensitive the result is to each measured angle, and choose the measuring position that minimises the error.

See a complete IA, marked

Our annotated exemplar When is the sea warm enough for our swimming club? A sinusoidal model of sea temperature (AI SL) asks a different question, but shows how a complete trig & navigation exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).