Room $8\times 6\times 4\text{ m}$. Find the floor diagonal.

1. Pythagoras : $d = \sqrt{64+36} = \mathbf{10\text{ m}}$.

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IB Maths AI SL · Unit 3: Geometry and Trigonometry

IB Maths AI SL 3D Trigonometry Questions

Updated October 2026

Exam-style IB Maths AI SL 3D trigonometry questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

13 questions
Paper 1: 13
5 easy
8 medium
3 worked examples

Practise 3D Trigonometry questions →

What you need to know

The single most common confusion in SL AI Paper 1. Sine rule when you have a matching angle-side pair; cosine rule when you have two sides + included angle. Practise until it's automatic. Sine rule vs cosine rule — the decision tree overview →

Standing at the top of a cliff looking down at a boat, or standing at the base of a tower looking up. SL AI always asks for BOTH the horizontal distance and the vertical height. 3D trigonometry — angle of elevation and depression overview →

Key formulas

Cosine rule: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$
Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

In the same notation as the IB formula booklet. All AI SL formulas →

Where students lose marks on 3D trigonometry

Examiner's noteTriangles and 3D: wrong rule, wrong angle, missing units

Where marks are lost: Candidates assumed a right angle that was not there (or used the sine rule in a right-angled triangle), found an angle other than the one named, used the wrong radius or only one of two areas, missed an arc length in a curved surface, forgot units where they were required, and did not give an answer in standard form when asked. Drawing a diagram helped.

Examiners flagged this in 3 exam series (May 2024, November 2024, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

3D Trigonometry worked examples

Worked example 1: Floor diagonal of a cuboid · easy

Room $8 \times 6 \times 4 m$ . Find the floor diagonal.

Solution

1. Pythagoras: $d = \sqrt{64 + 36} = 10 m$ .

Examiner tip: Sketch the 2-D triangle from a 3-D solid — 3D problems become 2D once you slice.

Worked example 2: Angle of space diagonal with the floor · medium

For the same $8 \times 6 \times 4$ room, find the angle the space diagonal makes with the floor.

Solution

3 steps

Examiner tip: Angle with base uses floor diagonal as adjacent, height as opposite.

Worked example 3: Slant-face angle of a pyramid · hard

Square pyramid: base side $10$ , height $12$ . Find the angle a slant FACE makes with the base.

Solution

3 steps

Examiner tip: Face-to-base uses HALF the SIDE. Edge-to-base uses HALF the DIAGONAL.

Try these IB Maths AI SL 3D trigonometry questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 2 marks · Paper 1

Calculate the exact distance between the 3D coordinates $A (1, 2, 4)$ and $B (4, - 2, 4)$ .

Attempt it and see the mark scheme →

Question 2 · medium · 3 marks · Paper 1

From the top of a vertical cliff $50$ m high, the angle of depression to a boat at sea is $24^{\circ}$ . Calculate the horizontal distance from the base of the cliff to the boat.

Attempt it and see the mark scheme →

Question 3 · medium · 3 marks · Paper 1

A square-based pyramid has a base of side length $6$ cm and a vertical height of $8$ cm. Calculate the angle between a sloping triangular face and the square base.

Full worked solution Free, no sign-in

Use the right-angled triangle from the centre of the base to the midpoint of a base edge: horizontal side $= \frac{6}{2} = 3$ cm, vertical side $= 8$ cm M1
Use the tangent ratio: $\tan θ = \frac{8}{3}$ M1
$θ = \tan^{- 1} (\frac{8}{3}) = {69.4}^{\circ}$ (3 s.f.) A1

Every question here has a full worked solution. See them all →

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All 13 3D trigonometry questions with mark schemes →

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FAQ

How many IB Maths AI SL 3D trigonometry questions are there?

There are 13 exam-style 3D trigonometry questions in the AI SL question bank (Paper 1: 13), graded 5 easy, 8 medium. Every question has a full IB-style mark scheme (M, A and R marks).

Is 3D trigonometry on Paper 1 or Paper 2?

In the question bank these questions are set as Paper 1 questions.

Where can I get the mark schemes?

Open the AI SL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Teaching this topic?

Free ready-to-teach lesson (slides and teacher notes): Right-angle trig applications.

IB Maths AI SL 3D Trigonometry Questions

Try it first

What you need to know

Key formulas

Where students lose marks on 3D trigonometry

3D Trigonometry worked examples

Worked example 1: Floor diagonal of a cuboid · easy

Solution

Worked example 2: Angle of space diagonal with the floor · medium

Solution

Worked example 3: Slant-face angle of a pyramid · hard

Solution

Try these IB Maths AI SL 3D trigonometry questions

Question 1 · easy · 2 marks · Paper 1

Question 2 · medium · 3 marks · Paper 1

Question 3 · medium · 3 marks · Paper 1

Full worked solution Free, no sign-in

Revise 3D trigonometry for the exam

FAQ

What else is free here

Teaching this topic?

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