3D Trigonometry

3D questions are ordinary 2D trigonometry — the difficulty is finding the right triangle inside the solid. Once you have drawn that triangle separately, everything you already know applies unchanged.

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1. The method

1. Identify the triangle you need. 2. REDRAW it separately, as a flat 2D triangle. 3. Label the sides you know, and mark the right angle. 4. Use Pythagoras or SOHCAHTOA.

Redrawing the triangle on its own is the single most useful step, and it was the technique tutors recommended most consistently here. Working inside the 3D picture is where students get lost — “which triangle to use” was the most frequently recorded difficulty.

Most 3D questions need TWO stages:

Stage 1: find a length in the base, usually with Pythagoras Stage 2: use that length in a vertical triangle

You often have to calculate a length before you can start. Assuming a diagonal equals a given edge — or guessing it — was a specific recorded error. If a side isn’t given, it must be calculated.

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2. The diagonal of a cuboid

For a cuboid with edges l, w, h:

Space diagonal: d² = l² + w² + h²

Where it comes from — two applications of Pythagoras:

1. The base diagonal: b² = l² + w²
2. The space diagonal: d² = b² + h² = l² + w² + h²

Example: a cuboid 3 × 4 × 12.

• Base diagonal = √(9 + 16) = 5
• Space diagonal = √(25 + 144) = √169 = 13

Deriving it in two steps is safer than recalling the formula, because it also gives you the base diagonal — which most angle questions then need.

Don’t round the base diagonal before using it. Carry the exact value into stage 2; rounding early was flagged repeatedly and loses accuracy marks.

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3. The angle between a line and a plane

This is the standard hard question.

The angle between a line and a plane is the angle between the line and its “shadow” — its projection — on that plane.

Method:

1. Identify where the line meets the plane
2. Drop a perpendicular from the top of the line to the plane
3. Join the foot of that perpendicular to the meeting point — this is the projection
4. The angle you want is between the line and its projection
5. That triangle has a right angle where the perpendicular meets the plane

Example: the angle between the space diagonal of a 3 × 4 × 12 cuboid and its base.

• The projection onto the base is the base diagonal, length 5
• The vertical height is 12
• tan θ = opposite/adjacent = 12/5
• θ = tan⁻¹(2.4) = 67.4° (3 s.f.)

Use the base diagonal, not an edge of the base. The projection of the space diagonal is the diagonal of the base — using 3 or 4 instead of 5 is the classic error.

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4. Pyramids and cones

For a square-based pyramid, the useful right-angled triangles are:

• Half the base diagonal, the vertical height, and the slant edge
• Half a base side, the vertical height, and the slant height of a face

Half. The apex sits above the centre of the base, so you use half the diagonal or half the side. Forgetting to halve is the standard slip.

For a cone: the radius, perpendicular height and slant height form a right-angled triangle: l² = r² + h².

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5. Choosing your method

Situation	Use
Two sides known, third wanted, right angle present	Pythagoras
Sides and an angle, right angle present	SOHCAHTOA
No right angle in that triangle	sine or cosine rule

Check for a right angle before reaching for the cosine rule. Using the cosine rule where SOHCAHTOA would do was recorded — it works, but it is slower and offers more chances to slip. Equally, don’t assume a right angle exists in a 3D figure just because the solid has square corners; the triangle you have drawn may not contain one.

The sine rule needs a known angle. Attempting it with no angles at all was a recorded error.

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6. Mistakes that cost marks

Not redrawing the triangle separately.

Choosing the wrong triangle.

Using an edge instead of the base diagonal for a projection.

Guessing a length instead of calculating it.

Forgetting to halve in a pyramid.

Rounding the stage-1 answer before stage 2.

Assuming a right angle that isn’t in your triangle.

Using the sine rule with no known angle.

Using tan where cos was needed (or vice versa) after mislabelling sides.

Leaving out a squared term in the cosine rule.

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Frequently asked questions

How do I approach a 3D trigonometry question? Find the triangle, redraw it flat, label it, then use Pythagoras or SOHCAHTOA.

How do I find the diagonal of a cuboid? d² = l² + w² + h², or apply Pythagoras twice.

What is the angle between a line and a plane? The angle between the line and its projection (its shadow) on the plane.

What is the projection of a cuboid’s space diagonal on the base? The base diagonal.

Why do I need two stages? Because the length you need for the vertical triangle usually has to be calculated in the base first.

Where is the apex of a square-based pyramid? Above the centre of the base — so use half the diagonal or half a side.

Should I round between stages? No — carry full accuracy and round only the final answer.

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Quick revision checklist

• I redraw the relevant triangle as a flat 2D diagram
• I label the sides and mark the right angle
• I expect two stages and plan them
• I can find the base diagonal with Pythagoras
• I know d² = l² + w² + h² and where it comes from
• I can identify a projection onto a plane
• I use the base diagonal, not an edge
• I halve the base diagonal or side for pyramids
• I check whether my triangle really has a right angle
• I choose between Pythagoras, SOHCAHTOA and the sine/cosine rule correctly
• I carry full accuracy between stages
• I give 3 s.f. and units

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These notes cover 3D trigonometry in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons. This was one of the less-covered topics in that set, so the page follows the syllabus closely rather than being padded. Always check the current syllabus and formula list for your own exam series.