Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

3-D Pythagoras & Trigonometry

In one sentence. Space diagonals, angles between lines and planes — finding the flat right-angled triangle hiding inside every solid. Higher Tier.

Trigonometry — one of 4 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Claudius Ptolemy. Learn 3-D Pythagoras & Trigonometry, teach it, then defend it against Claudius Ptolemy.

Where this sits in the specification

Specification 4MA1, 4.8 Pythagoras and trigonometry in three dimensions (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 4.8 — Pythagoras and trigonometry in three dimensions (Higher)

1 · Understand it

What it is

3-D problems are 2-D problems in disguise: every length or angle you need lives in some FLAT right-angled triangle inside the solid. Space diagonals need Pythagoras twice (base diagonal, then up the height). The angle between a line and a plane is measured against the line's shadow (projection) on that plane.

Why it works

A cuboid's space diagonal, its base diagonal and its vertical edge form a genuine right triangle — the vertical edge is perpendicular to the whole base plane, so the right angle is guaranteed. That's the general trick: use a vertical edge to manufacture a right angle, and the 2-D toolkit applies unchanged.

When a question is asking for it

Longest-rod-in-a-box questions; pyramid slant edges and slant faces; 'the angle between the diagonal and the base'. Strategy is always the same: choose the triangle, DRAW it flat with its three labels, then run 2-D Pythagoras or trig on it.

How to hold it in your head

The angle a leaning pole makes with the ground is the angle with its own SHADOW at noon. Height is opposite, shadow is adjacent — tan θ = height/shadow. If you can identify the shadow, you've found the triangle.

2 · The mistakes that cost marks

  • Doing 3-D in one leap. Two clean 2-D steps beat one confused 3-D one — find the base diagonal first.
  • Measuring the angle with a plane against an EDGE instead of the projection (shadow) of the line.
  • Using the full base diagonal where the triangle needs half (pyramid apex questions measure from the CENTRE).
  • Rounding the intermediate diagonal, then compounding the error in step two. Carry exact or full-precision values.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

A cuboid has a 6 cm × 8 cm base and height 24 cm. Find its space diagonal.

  1. Base diagonal: √(6² + 8²) = 10.
  2. Space diagonal: √(10² + 24²) = √676 = 26 cm.

Answer: 26 cm

A square-based pyramid has base side 10 cm and apex 12 cm above the base's centre. Find a slant edge.

  1. Centre to a base corner: half the base diagonal = (10√2)/2 = 5√2.
  2. Slant edge: √((5√2)² + 12²) = √(50 + 144) = √194.
  3. = 13.93 cm (2 d.p.).

Answer: 13.93 cm

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — A cuboid has a base measuring 6 cm by 8 cm. Work out the length of the base's diagonal. (answer in cm)
  2. Grade 7 — The same cuboid (base 6 cm × 8 cm) has height 24 cm. Work out the length of the space diagonal (corner to opposite corner through the inside). (answer in cm)
  3. Grade 8 — For the same cuboid, work out the angle between the space diagonal and the BASE, correct to 1 decimal place. (answer in °)
  4. Grade 9 — A square-based pyramid has base side 10 cm, and its apex is 12 cm vertically above the CENTRE of the base. Work out the length of a slant edge (apex to a base corner), correct to 2 decimal places. (answer in cm)
  5. 9+ — For the same pyramid, work out the angle between a slant edge and the base, correct to 1 decimal place. (answer in °)

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on 3-D Pythagoras & Trigonometry, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how 3-D Pythagoras & Trigonometry works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue 3-D Pythagoras & Trigonometry against Claudius Ptolemy — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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