A rectangular room has dimensions: length $8 \text{ m}$, width $5 \text{ m}$, and height $4 \text{ m}$. Calculate the exact length of the space diagonal (the straight line from one bottom corner to the opposite top corner). Calculate the angle this space diagonal makes with the floor of the room.

Diagonal $= \sqrt{8^2 + 5^2 + 4^2}$ M1 $= \sqrt{64 + 25 + 16} = \sqrt{105} \text{ m}$ (or $\approx 10.2 \text{ m}$). A1 Base diagonal $= \sqrt{8^2 + 5^2} = \sqrt{89}$. M1 $\tan \alpha = \frac{4}{\sqrt{89}} \implies \alpha = \tan^{-1}(4/\sqrt{89}) \approx 23.0^{\circ}$. A1

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IB Maths AI SL · Unit 3: Geometry and Trigonometry

IB Maths AI SL 3D Coordinate Geometry Questions

Updated October 2026

Exam-style IB Maths AI SL 3D coordinate geometry 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: 11
Paper 2: 2
3 easy
6 medium
2 hard
2 very hard
3 worked examples

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What you need to know

SL AI's geometry is applied: which cereal box has the lowest surface-area-to-volume ratio? Master the volume + surface-area formulas for cylinders, cones, and truncated shapes. 3D geometry — volume, surface area, and packaging overview →

3D Coordinate Geometry worked examples

Worked example 1: Distance between two points in 3D · easy

Find the exact distance between $A (1, 4, - 2)$ and $B (3, 5, 2)$ .

Solution

4 steps

Examiner tip: Take care with negatives:

2 - (- 2) = 4

. Parenthesise before squaring.

Worked example 2: Midpoint of a 3D flight path · medium

A drone flies from $P (- 2, 10, 5)$ to $Q (6, - 4, 13)$ . Find the exact midpoint.

Solution

5 steps · 1 to fill in

Examiner tip: Midpoints ADD then divide by 2. Distance formula SUBTRACTS.

Worked example 3: Missing z-coordinate from a given distance · hard

The distance between $C (0, 1, - 1)$ and $D (2, 3, z)$ is exactly $3$ units. Find both values of $z$ .

Solution

5 steps

Examiner tip: Always write

\pm

when square-rooting both sides — missing the negative root loses half the marks.

Try these IB Maths AI SL 3D coordinate geometry 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 · 3 marks · Paper 1

The points $P (- 1, 3, 5)$ and $Q (5, 7, - 1)$ represent the endpoints of a tunnel in a 3D grid.

Find the exact coordinates of the midpoint of the tunnel.
Find the exact distance between point P and point Q.

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Question 2 · medium · 4 marks · Paper 2

A rectangular room has dimensions: length $8 m$ , width $5 m$ , and height $4 m$ .

Calculate the exact length of the space diagonal (the straight line from one bottom corner to the opposite top corner).
Calculate the angle this space diagonal makes with the floor of the room.

Full worked solution Free, no sign-in

Diagonal $= \sqrt{8^{2} + 5^{2} + 4^{2}}$ M1
$= \sqrt{64 + 25 + 16} = \sqrt{105} m$ (or $\approx 10.2 m$ ). A1
Base diagonal $= \sqrt{8^{2} + 5^{2}} = \sqrt{89}$ . M1
$\tan α = \frac{4}{\sqrt{89}} ⟹ α = \tan^{- 1} (4 / \sqrt{89}) \approx {23.0}^{\circ}$ . A1

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

Question 3 · hard · 4 marks · Paper 1

A rectangular cuboid sits on a 3D coordinate system. Its vertices include the origin $O (0, 0, 0)$ , and the points $A (5, 0, 0)$ , $B (0, 4, 0)$ , and $C (0, 0, 3)$ . Let point $D$ be the vertex diagonally opposite to the origin.

State the coordinates of point $D$ .
Find the exact length of the space diagonal $O D$ .
Find the exact coordinates of the midpoint of $O D$ .

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FAQ

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

There are 13 exam-style 3D coordinate geometry questions in the AI SL question bank (Paper 1: 11 · Paper 2: 2), graded 2 very hard, 2 hard, 6 medium, 3 easy. Every question has a full IB-style mark scheme (M, A and R marks).

Is 3D coordinate geometry on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 11 · Paper 2: 2. Practise with your GDC — AI papers expect calculator methods throughout.

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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Free ready-to-teach lesson (slides and teacher notes): Distance & midpoint in 3D.

IB Maths AI SL 3D Coordinate Geometry Questions

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What you need to know

3D Coordinate Geometry worked examples

Worked example 1: Distance between two points in 3D · easy

Solution

Worked example 2: Midpoint of a 3D flight path · medium

Solution

Worked example 3: Missing z-coordinate from a given distance · hard

Solution

Try these IB Maths AI SL 3D coordinate geometry questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 4 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 4 marks · Paper 1

Revise 3D coordinate geometry for the exam

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