IB Maths AI SL · Unit 3: Geometry and Trigonometry
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
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 →
Room $8\times 6\times 4\text{ m}$. Find the floor diagonal.
1. Pythagoras: $d = \sqrt{64+36} = \mathbf{10\text{ m}}$.
For the same $8\times 6\times 4$ room, find the angle the space diagonal makes with the floor.
1. Right triangle: floor diagonal $= 10$, height $= 4$.
2. Ratio: $\tan\theta = \frac{4}{10}$.
3. State: $\theta = \mathbf{21.8^\circ}$.
Square pyramid: base side $10$, height $12$. Find the angle a slant FACE makes with the base.
1. Apothem: half the base side $= 5$.
2. Ratio: $\tan\theta = \frac{12}{5}$.
3. State: $\theta = \mathbf{67.4^\circ}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Calculate the exact distance between the 3D coordinates $A(1, 2, 4)$ and $B(4, -2, 4)$.
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.
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.
There are 12 exam-style 3d trigonometry questions in the AI SL question bank (Paper 1: 12), graded 4 easy, 8 medium. Every question has a full IB-style mark scheme (M, A and R marks).
In the question bank these questions are set as Paper 1 questions.
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.