Sheet № 51 · Higher only · AQA · Edexcel · OCR
Trigonometry in 3D –
Trigonometry in 3D is a Higher tier topic that builds on everything you already know about right-angled triangle trigonometry and Pythagoras' theorem, but applies it to three-dimensional shapes such as cuboids, pyramids, and prisms. Exam boards — AQA, Edexcel, and OCR — regularly test your ability to identify right-angled triangles hidden
§Key definitions
Question:
A cuboid has length 8 cm, width 6 cm, and height 5 cm. Calculate the length of the space diagonal AG. Then find the angle that AG makes with the base ABCD. Give your answers to 1 decimal place.
Answer:
The space diagonal is 11.2 cm and the angle with the base is 26.6°.
Question 1:
A cuboid measures 12 cm by 5 cm by 4 cm. Find the length of the space diagonal. Give your answer to 1 decimal place.
Question 2:
A cuboid has length 10 cm, width 8 cm, and height 6 cm. Find the angle the space diagonal makes with the base of the cuboid. Give your answer to 1 decimal place.
Question 3:
A square-based pyramid has a base of side 6 cm and a slant height of 10 cm (from the midpoint of a base edge to the apex). Find the vertical height of the pyramid. Give your answer to 1 decimal place.
§Formulas to memorise
Pythagoras in 3D — For a cuboid with sides a, b, and c the space diagonal d = sqrt(a² + b² + c²)
SOHCAHTOA — sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
Pythagoras in 2D — a² + b² = c² (used as a stepping stone inside 3D problems)
Draw and label the 3D shape. — Mark all given lengths clearly.
Identify the right-angled triangle you need. — This usually involves a diagonal across a face (found first using 2D Pythagoras) and then a second triangle using that diagonal as one side.
Extract the triangle — redraw it as a flat 2D triangle with all known sides labelled.
Apply Pythagoras or SOHCAHTOA — to find the unknown length or angle.
Give your answer to the required degree of accuracy — (usually 3 significant figures or 1 decimal place).
Use tan θ = opposite / adjacent (or another ratio if you prefer).
Worked example
A cuboid has length 8 cm, width 6 cm, and height 5 cm. Calculate the length of the space diagonal AG. Then find the angle that AG makes with the base ABCD. Give your answers to 1 decimal place.
Working:
⚠ Common mistakes
- ✗Not using a two-step approach. Many students try to jump straight to the space diagonal without first finding the diagonal across a face. Always work in stages.
- ✗Mixing up opposite and adjacent. When finding the angle between a line and a plane, the perpendicular height is always the opposite side. Double-check by asking: "Which side is across from the angle?"
- ✗Rounding too early. Keep intermediate values unrounded (use the full calculator display) and only round your final answer. Early rounding causes inaccurate results.
- ✗Forgetting to square root. After using Pythagoras, remember that you have found c² — you must take the square root to get c.
- ✗Not labelling the triangle clearly. If you extract the triangle but do not label which sides correspond to which parts of the 3D shape, you risk using the wrong values.
✦ Exam tips
- →Always sketch the 2D triangle separately. This makes it much easier to apply SOHCAHTOA correctly and earns method marks even if your final answer is wrong.
- →Show every step. In 3D trig questions (usually worth 4–5 marks), examiners award marks for identifying the correct triangle, setting up the calculation, and arriving at the answer.
- →State the trig ratio you are using (e.g. "Using tan θ = opp / adj") before substituting values. This picks up method marks.
- →Use the ANS button on your calculator to avoid copying long decimals — this prevents rounding errors.
- →Check your answer makes sense. An angle between a line and a base should be between 0° and 90°. A space diagonal should be longer than any single edge.