What is the first step in a 3D trigonometry problem?

find a right-angled triangle and draw it flat. Then it is an ordinary 2D problem.

The angle between a line and a plane is measured between the line and what?

its projection (shadow) on the plane. The projection is directly underneath the line.

What is the formula for the longest diagonal of a cuboid with sides a, b and c?

√(a² + b² + c²). It is Pythagoras used twice.

Pythagoras and trigonometry in 3D · Cambridge IGCSE Mathematics

In plain words

Three-dimensional problems look harder than they are. The method is always the same: find a right-angled triangle hiding inside the solid, draw it flat on its own, and use Pythagoras or SOH CAH TOA as usual.

Three things to know

Find a right-angled triangle that contains the length or angle you want. Draw it separately and label it.
The longest diagonal of a cuboid with sides a, b and c is √(a² + b² + c²).
The angle between a line and a plane is the angle between the line and its "shadow" on the plane.

Worked example

A cuboid measures 3 cm by 4 cm by 12 cm. Find the length of its longest diagonal.

First the diagonal of the base: √(3² + 4²) = 5 cm.
That diagonal and the 12 cm height make a right-angled triangle with the long diagonal.
√(5² + 12²) = √169 = 13 cm.

Tips and tricks

Draw the triangle out flat every time. Trying to do it on the 3D picture is how mistakes happen.
Keep unrounded values in your calculator between steps.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Pythagoras and trigonometry in 3D: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the longest diagonal of a cuboid 2 cm by 3 cm by 6 cm?
- 11 cm
- 7 cm (the answer)
- 6.3 cm
- 49 cm
√(4 + 9 + 36) = √49.
What is the first step in a 3D trigonometry problem?
- measure the diagram
- find a right-angled triangle and draw it flat (the answer)
- use the sine rule
- find the volume
Then it is an ordinary 2D problem.
What is the diagonal of the base of a cuboid whose base is 6 cm by 8 cm?
- 14 cm
- 10 cm (the answer)
- 48 cm
- 7 cm
√(36 + 64).
The angle between a line and a plane is measured between the line and what?
- the vertical
- the nearest edge
- its projection (shadow) on the plane (the answer)
- the longest side
The projection is directly underneath the line.
What is the formula for the longest diagonal of a cuboid with sides a, b and c?
- a + b + c
- √(a² + b² + c²) (the answer)
- abc
- √(a² + b²) + c
It is Pythagoras used twice.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 9 marks. Write your answers on paper, then check them.

Pythagoras and trigonometry in 3D

Cambridge IGCSE Mathematics 0580 · 9 marks · papermunch.org

Name ______________________________ Date ______________

Find the longest diagonal of a cube with edges of 5 cm.[3]
Show answerHide answer
8.66 cm. √75.
I got this one
A vertical pole 6 m high stands at one corner of a horizontal rectangle 3 m by 4 m. Find the distance from the top of the pole to the opposite corner of the rectangle.[3]
Show answerHide answer
7.81 m. The diagonal of the rectangle is 5 m; √(25 + 36).
I got this one
A cuboid has a base diagonal of 10 cm and a height of 5 cm. Find the angle between its longest diagonal and the base.[3]
Show answerHide answer
26.6°. tan⁻¹(5 ÷ 10).
I got this one

Answers: Pythagoras and trigonometry in 3D

1. 8.66 cm. √75.
2. 7.81 m. The diagonal of the rectangle is 5 m; √(25 + 36).
3. 26.6°. tan⁻¹(5 ÷ 10).