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MathidiaMathematics Academy
Advanced geometry and solids
Lesson 51 of 75
Introduction

3D Pythagoras and trigonometry

Objective. Construct linked right triangles inside solids.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

3D Pythagoras and trigonometry finds lengths and angles inside solids by slicing them into flat right-angled triangles. First spot a right-angled triangle hidden in the solid, often using a diagonal of the base, then apply Pythagoras or SOHCAHTOA to that flat triangle. For a cuboid with edges 3, 4, 12, the space diagonal has length √(3²+4²+12²) = √169 = 13. Angles between a line and a plane are found by dropping a perpendicular to the plane and using the triangle formed with the projection. Always redraw the relevant 2D triangle separately before calculating.

Worked example

A cuboid has length 6 cm, width 2 cm and height 3 cm. Find the length of its space diagonal.

Diagram of a cube
  1. 1Space diagonal d satisfies d² = 6² + 2² + 3²
  2. 2d² = 36 + 4 + 9 = 49
  3. 3d = √49
  4. 4d = 7 cm

Answer: 7 cm

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

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Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

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Common mistake. A common error is using the wrong diagonal, such as a face diagonal instead of the space diagonal, giving an undersized answer. Always check whether all three dimensions of the solid are genuinely involved in the triangle chosen.

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