Reduce 3D to 2D: In 3D trig, you never solve the whole 3D problem at once. You identify a right triangle embedded in the 3D shape, extract it, label its sides, and solve in 2D. The key skill is finding the right triangle.
| 3D shape | Right triangle to use | Key lengths |
|---|---|---|
| Cuboid | Body diagonal triangle: base diagonal + height | base diag = √(l²+w²) |
| Pyramid (square base) | Triangle from apex to base midpoint | height + half base diagonal |
| Cuboid | Face diagonal on any face | Use Pythagoras on that face |
Space diagonal formula: The space diagonal of a cuboid with dimensions l × w × h goes from one corner to the opposite corner through the interior.
Worked example — space diagonal
A cuboid is 6 cm × 4 cm × 3 cm. Find the length of the space diagonal and the angle it makes with the base.
Step by step
- Space diagonal d.
- Base diagonal (the adjacent side in the angle triangle).
- Angle θ with the base: tan θ = height / base diagonal.
Final answer
Space diagonal ≈ 7.81 cm; angle with base ≈ 22.6°.
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Worked example — angle at apex of pyramid
A square pyramid has base side 10 cm and height 12 cm. Find the angle between a slant edge and the base.
Step by step
- The slant edge runs from the apex to a base corner. Find half the base diagonal first.
- The right triangle has leg: height 12 cm and horizontal distance 5√2 cm.
- Angle at the base.
Final answer
Slant edge ≈ 13.93 cm; angle with base ≈ 59.5°.
Worked example — angle between two lines in a cuboid
A room is a cuboid 8 m long, 5 m wide, and 3 m high. Find the angle between the space diagonal and the longest edge.
Step by step
- Space diagonal d = √(8²+5²+3²) = √(64+25+9) = √98 ≈ 9.90 m.
- Angle with the 8 m edge: cos θ = 8/d.
Final answer
Angle ≈ 36.2°.
IB mark allocation hint: In Paper 2, 3D trig questions are usually worth 3–6 mark. Show each sub-triangle clearly on your diagram to earn method marks even if you mis-calculate.