Trigonometry
Grade 8-93D Trigonometry & Pythagoras
Every 3D trigonometry problem reduces to a sequence of ordinary right-angled triangles hidden inside the solid, and the skill is spotting which flat triangle to work with at each stage. This lesson covers finding the diagonal of a cuboid, finding the angle between a diagonal and the base, and finding the height and slant-edge angle of a right pyramid with a square base.
What you need to know
- Pick a right-angled triangle inside the 3D shape and redraw it flat as a 2D diagram.
- You often need Pythagoras or trig on one triangle to get a length for the next.
- The angle between a line and a plane is measured to the line shadow on that plane.
- Work in stages and keep full accuracy until the final answer.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Finding the diagonal of a cuboid
The diagonal running from one corner of a cuboid to the opposite corner can be found in one step using a 3D version of Pythagoras: d² = length² + width² + height², combining all three edge lengths at once.
A cuboid measuring 6 cm by 4 cm by 3 cm has a diagonal of √(6² + 4² + 3²) = √61 = 7.81 cm (3 s.f.).
Finding the angle between a diagonal and the base
First find the diagonal of the base rectangle using ordinary 2D Pythagoras. This base diagonal, together with the vertical height, forms a right-angled triangle with the main diagonal as its hypotenuse, so tan(angle) = height ÷ base diagonal.
A cuboid has a base of 8 cm by 6 cm and a height of 5 cm. The base diagonal is √(8² + 6²) = 10 cm. The angle between the main diagonal and the base is tan⁻¹(5 ÷ 10) = 26.6° (1 d.p.).
Finding the height of a right pyramid
For a right pyramid with a square base, the height, half of the base diagonal, and the slant edge form a right-angled triangle, with the slant edge as the hypotenuse. Find the base diagonal with 2D Pythagoras, halve it, then use Pythagoras again to find the height.
A pyramid has a square base of side 6 cm and a slant edge of 9 cm. The base diagonal is √(6² + 6²) = √72, so half the diagonal squared is 72 ÷ 4 = 18. The height is √(9² - 18) = √63 = 7.94 cm (3 s.f.).
Finding the angle between a slant edge and the base
The angle between a slant edge and the base is found using the same right-angled triangle as the height, with tan(angle) = height ÷ half the base diagonal.
A pyramid has a square base of side 10 cm and a height of 12 cm. Half the base diagonal is √(10² + 10²) ÷ 2 = 7.07 cm. The angle between the slant edge and the base is tan⁻¹(12 ÷ 7.07) = 59.5° (1 d.p.).
Worked examples
Three exam-style questions, fully solved
A cuboid measures 6 cm by 4 cm by 3 cm. Find the length of the diagonal AG running from one corner to the opposite corner. Give your answer correct to 3 significant figures.
Easy- 1.Use the 3D Pythagoras formula: AG² = 6² + 4² + 3²
- 2.Calculate: AG² = 36 + 16 + 9 = 61
- 3.Take the square root: AG = √61
Answer: 7.81 cm
A cuboid has a base of 8 cm by 6 cm and a height of 5 cm. Find the angle between the diagonal AG and the base ABCD (angle GAC). Give your answer correct to 1 decimal place.
Medium- 1.Find the base diagonal using 2D Pythagoras: AC² = 8² + 6² = 100, so AC = 10 cm
- 2.Set up the trig ratio using the height and base diagonal: tan(angle GAC) = 5 ÷ 10
- 3.Use the inverse tangent: angle GAC = tan⁻¹(0.5)
Answer: 26.6°
A right pyramid has a square base of side 10 cm and a height VM of 12 cm. Find the angle between the slant edge VA and the base (angle VAM). Give your answer correct to 1 decimal place.
Hard- 1.Find the base diagonal: diagonal² = 10² + 10² = 200
- 2.Halve the diagonal to find AM: AM² = 200 ÷ 4 = 50, so AM = 7.07 cm (3 s.f.)
- 3.Set up the trig ratio using the height and AM: tan(angle VAM) = 12 ÷ 7.07
Answer: 59.5°
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
A cuboid measures 3 cm by 4 cm by 12 cm. Work out the length of the longest diagonal (from one corner to the opposite corner).
A cuboid measures 2 cm by 6 cm by 9 cm. Work out the length of its longest diagonal.
A cuboid has a rectangular base 8 cm by 6 cm and a height of 5 cm. Work out the length of the diagonal of the base.
Using the base diagonal from the previous question (10 cm) and the height of 5 cm, work out the length of the space diagonal of that cuboid, to 1 decimal place.
In a cuboid, the space diagonal makes an angle θ with the base. The base diagonal is 10 cm and the height is 6 cm. Work out θ, to 1 decimal place.
A cube has edges of 5 cm. Work out the length of the space diagonal, to 1 decimal place.
Avoid these
The mistakes students make most often
Trying to find a cuboid diagonal in a single step without applying Pythagoras twice (or the combined 3D formula), missing one of the three dimensions.
Using the full base diagonal instead of half of it when finding the height or slant-edge angle of a pyramid, since M is the centre of the base.
Using the wrong two sides in the trig ratio when finding an angle, for example the base diagonal instead of the height, or the slant edge instead of the base distance.
Rounding an intermediate value, such as the base diagonal or half-diagonal, too early, which throws off the accuracy of the final answer.
Confusing which angle is being asked for, for example finding the angle with a vertical edge instead of the angle between the diagonal and the base.
FAQ
Questions students and parents ask
Before this topic, make sure you know
What to learn next
Teachably is a GCSE Maths tutoring practice
This hub is our free resource and it stays free. If you would like one-to-one help alongside it, a short call is the best place to start. There is also a free video course for parents.