Ch 3 · 6.4 Pythagoras' Theorem and Trigonometry in 3D · O Level Mathematics 4024
Topic 3 of 3 in Trigonometry — O Level Mathematics 4024 — 18 lessons.
Introduction
How long a pole fits diagonally in a box? How steep is the edge of a pyramid? Three-dimensional problems are solved by finding right-angled triangles inside the solid and using the 2D tools on each in turn. (Syllabus section 6.4; in IGCSE 0580 it is section E6.6.)
Key ideas
Sketch the solid, then redraw each useful triangle flat, with its right angle marked. Typical triangles: a face diagonal on the base, and a vertical edge rising from it.
🧑🏫 Teacher's explanation
Imagine you are standing in the corner of a rectangular room. If you want to walk from the bottom corner to the opposite top corner, you aren't just moving across the floor or up the wall; you are moving through the 'space' of the room. To find this distance, you can't just use 2D math; you have to combine the floor distance and the height using Pythagoras' theorem twice.
📖 Definitions
🔍 Worked example — Easy
A cuboid has dimensions 3 cm × 4 cm × 12 cm. Calculate the length of the space diagonal.
🔍 Worked example — Standard
A pole of height 5 m is leaning against a vertical wall. The foot of the pole is 3 m from the base of the wall. Find the angle θ that the pole makes with the ground.
🔍 Worked example — Exam-style
A rectangular pyramid has a square base of side 10 cm. The vertical height of the pyramid is 12 cm. Calculate the angle between one of the sloping edges and the base of the pyramid.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
A cuboid is 3 cm by 4 cm by 12 cm high. Base diagonal = 5 cm; space diagonal = √(9 + 16 + 144) = 13 cm. Angle between the space diagonal and the base = tan^-1 (12/5 ) = 67.4^°.
🔍 Worked example — Easy
A cuboid has dimensions 3 cm × 4 cm × 12 cm. Calculate the length of the space diagonal and the angle it makes with the base.
🔍 Worked example — Standard
A square-based pyramid has a base of 8 cm × 8 cm. The vertical height from the centre of the base to the apex is 10 cm. Find the length of a sloping edge.
🔍 Worked example — Exam-style
A line segment AB starts at the origin (0,0,0) and ends at point P(4, 3, 12). Calculate the angle between the line AP and the xy-plane.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) √(900 + 1600 + 2500) = 70.7 cm. (2) base diagonal 50 cm; tan^-150/50 = 45^°. (3) diagonal 50 m; tan^-120/50 = 21.8^°.