This topic summary covers Knowledge Organiser: Pythagoras in 3D within Pythagoras in 3D for GCSE Mathematics. Revise Pythagoras in 3D in Geometry & Measures for GCSE Mathematics with 12 exam-style questions and 2 flashcards. Use this page as part of a wider topic revision path rather than treating it as an isolated fact. It is section 5 of 5 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Pythagoras in 3D
Key Terms
- Space diagonal: The longest diagonal through a 3D shape
- Base diagonal: The diagonal across the base face
- Perpendicular height: The height at 90° to the base
- Cuboid: A 3D rectangle with three different dimensions x, y, z
Must-Know Facts
- Every 3D Pythagoras problem hides right-angled triangles — find and draw them
- Apply Pythagoras TWICE for a space diagonal (step on base, then step up)
- Space diagonal direct formula: d² = x² + y² + z²
- Slant height of cone: l = √(r² + h²)
- Always draw a clear 2D right-angled triangle extracted from the 3D shape
Key Formulas
- Space diagonal: d = √(x² + y² + z²)
- Step 1 (base): b = √(x² + y²)
- Step 2 (space): d = √(b² + z²)
- Cone slant height: l = √(r² + h²)
Common Mistakes
- Doing it in one step: Must use two applications of Pythagoras — find the base diagonal first, then use it for the space diagonal
- Using slant vs vertical height: The space diagonal of a cuboid uses all three dimensions; don't confuse with a face diagonal
- Not identifying the right triangle: Sketch the 3D shape and clearly mark the right-angled triangles before calculating
- Rounding intermediate answers: Keep full accuracy until the final step to avoid accumulated errors
Practice questions for Pythagoras in 3D
A cuboid has length l, width w, and height h. Which formula gives the length of the space diagonal d?
Describe the two-step method for finding the space diagonal of a cuboid with dimensions l, w, and h.
Quick recall flashcards
3D Pythagoras
d² = x² + y² + z² for space diagonal. Or use 2D Pythagoras twice: first on base, then with height.
3D Pythagoras
Use Pythagoras twice: once for base diagonal, once for 3D diagonal