What you'll learn
This topic covers trigonometry in 3D — finding lengths and angles inside solids such as cuboids and pyramids. In this guide you will learn how to spot right-angled triangles inside 3D shapes, how to find the angle between a line and a plane, how to use Pythagoras and trigonometry together, and how to work step by step through 3D problems. These are challenging higher-tier skills.
Key terms and definitions
Plane — a flat 2D surface within a 3D shape.
Angle between a line and a plane — the angle the line makes with its "shadow" on the plane.
Space diagonal — the longest diagonal across a solid.
Right-angled triangle — the 2D triangle you extract to apply trigonometry.
Foot of the perpendicular — where a vertical line meets the base plane.
Core concepts
Finding right-angled triangles in 3D
The key skill is to identify a right-angled triangle inside the solid that contains the length or angle you want. Sketch that triangle separately as a flat 2D shape, label its sides, and then use Pythagoras or SOH CAH TOA on it.
Using Pythagoras twice
To find a space diagonal, often you use Pythagoras twice: first find a diagonal across the base, then combine it with the height in a second right-angled triangle. Alternatively use the 3D formula d = √(a² + b² + c²) for a cuboid.
Angle between a line and a plane
The angle between a line and a plane is found by dropping a perpendicular from the line's end to the plane, forming a right-angled triangle. The angle is between the line and its projection (shadow) on the plane. Identify the right angle at the foot of the perpendicular.
Working step by step
3D problems usually need two or more steps: find an intermediate length first (often a base diagonal), then use it in the next triangle. Keep full accuracy for the intermediate value to avoid rounding errors.
Combining tools
You combine Pythagoras (for lengths) and trigonometry (for angles) within the same problem. Decide which triangle and which tool each step needs, and label clearly so you do not lose track.
Worked examples
Example 1: Base diagonal
Find the diagonal of a rectangular base 6 by 8.
√(6² + 8²) = √100 = 10.
Example 2: Space diagonal
Find the space diagonal of a cuboid 2 by 3 by 6.
√(2² + 3² + 6²) = √49 = 7.
Example 3: Angle to the base
A vertical post of height 5 stands at the corner of a base; the diagonal to its top has base part 10. Find the angle to the base.
tan θ = 5/10, θ = tan⁻¹(0.5) = 26.6° (3 s.f.).
Common mistakes and how to avoid them
Not extracting a 2D triangle. Always redraw the right-angled triangle flat.
Using the wrong angle. The line–plane angle is with the projection on the plane.
Rounding the intermediate length. Keep full accuracy for the next step.
Forgetting Pythagoras twice. Find the base diagonal before the space diagonal.
Mislabelling sides. Identify hypotenuse, opposite and adjacent for each triangle.
Exam technique for 3D Trigonometry
Identify and redraw the right-angled triangle in 2D.
Use Pythagoras twice (or the 3D formula) for space diagonals.
Drop a perpendicular to find the angle between a line and a plane.
Keep full accuracy for intermediate values.
Combine Pythagoras and trigonometry step by step.
Quick revision summary
3D trigonometry works by finding a right-angled triangle inside the solid and redrawing it flat as a 2D triangle, then applying Pythagoras or SOH CAH TOA. For a space diagonal, use Pythagoras twice — first a base diagonal, then combine it with the height — or the cuboid formula d = √(a² + b² + c²). The angle between a line and a plane is found by dropping a perpendicular to the plane and measuring the angle between the line and its projection (shadow), with the right angle at the foot of the perpendicular. Problems take several steps, so find an intermediate length first and keep full accuracy before the next step, combining Pythagoras (lengths) and trigonometry (angles). The common errors are not extracting a 2D triangle, using the wrong angle, rounding intermediate values, and forgetting to apply Pythagoras twice. Redraw the triangle, use Pythagoras twice for diagonals, drop perpendiculars for line–plane angles, and keep full accuracy throughout.