Key Terms
- Space diagonal: The longest diagonal through a 3D solid
- Angle of elevation: Angle measured upward from horizontal
- Angle of depression: Angle measured downward from horizontal
- Right-angled triangle in 3D: A 2D triangle extracted from a 3D shape
Must-Know Facts
- Every 3D problem is really a series of 2D problems — extract the right triangles
- Draw each 2D right-angled triangle separately and label sides clearly
- Often need to find the base diagonal first as a stepping stone
- Use Pythagoras for distances; use SOHCAHTOA for angles
- Space diagonal: d = √(x² + y² + z²)
Key Methods
- Angle with base: tan θ = height ÷ base diagonal
- Space diagonal: d = √(x² + y² + z²)
- Step 1: find base diagonal; Step 2: use with height
- Always state which triangle you are working with
Key Formulas
- Space diagonal: d = √(l² + w² + h²)
- Angle with base: tan θ = h ÷ base diagonal
- Base diagonal (cuboid): √(l² + w²)
- Same SOHCAHTOA rules apply — identify the right-angled triangle first
Common Mistakes
- Not finding base diagonal first: Must apply Pythagoras twice — base diagonal then space diagonal/angle
- Using wrong triangle: Sketch the 3D shape and clearly identify the right-angled triangle being used
- Rounding intermediate values: Keep full calculator accuracy until the final answer to avoid compounding errors
- Angle with wrong side: The angle with the base uses the perpendicular height and the base diagonal — not a slant edge
This topic summary covers Knowledge Organiser: 3D Trigonometry within 3D Trigonometry for GCSE Mathematics. Revise 3D Trigonometry 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 6 of 6 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for 3D Trigonometry
To find the angle between a line and a horizontal plane in a 3D problem, which technique is typically used?
Describe the general method for finding the angle between a line and a plane in a 3D problem.
Quick recall flashcards
Key Strategy for 3D Trig
Find the right-angled triangles hidden in the 3D shape. Often you need to find the BASE DIAGONAL first!
3D Pythagoras Formula
d² = x² + y² + z² for a space diagonal through a cuboid with dimensions x, y, z