Three-dimensional trigonometry

Objective. Identify hidden right triangles in prisms and pyramids.

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2. Worked example
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Learn: the key idea

Three-dimensional trigonometry means spotting a flat right-angled triangle hidden inside a 3D shape like a cuboid, prism or pyramid, then solving it with ordinary trig or Pythagoras. First identify the relevant 2D cross-section, redraw it flat, label known lengths and angles, then apply SOH-CAH-TOA. For example, to find the angle between a cuboid's space diagonal and its base, use the base diagonal and the vertical height as two sides of a right triangle. Always sketch the 2D triangle separately so 3D clutter doesn't obscure which sides are which.

Worked example

A cuboid has base 6 cm by 8 cm and height 5 cm. Find the angle between the diagonal of the base and the vertical edge forming a right-angled triangle, through three-dimensional trigonometry, to 1 decimal place.

1 $Base diagonal = \sqrt(6²+8²) = \sqrt100 = 10 cm$
2 $Right triangle has base diagonal 10 cm and height 5 cm$
3 $tan θ = 10/5 = 2 (angle between diagonal and vertical edge)$
4 $θ = tan⁻¹(2) \approx 63.4°$

Answer: 63.4°

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Common mistake. Students often use the full 3D space diagonal when the base diagonal is actually the side needed; carefully extract the exact right-angled triangle being asked about before applying trigonometry.

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