IGCSE Math RevisionQuestion 1
A cuboid measures \(3\) cm by \(4\) cm by \(12\) cm.
Find the length of its longest diagonal.
Show the worked answer
- \(d = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13\) cm
Answer: \(13\) cm
Cambridge IGCSE Maths (0580) · Topic 6: Trigonometry · 6.6
Cambridge IGCSE Maths 0580 3D Pythagoras and Trigonometry Questions
3D Pythagoras and Trigonometry questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.
- 8 questions
- 8 Extended
- 0 non-calculator
- Free, no sign-in
What the syllabus asks
- 6.6 Pythagoras' theorem and trigonometry in 3D Extended only
Extended: Pythagoras and trigonometry in three dimensions, including the angle between a line and a plane.
Summarised in our own words: check the exact wording in the official 0580 syllabus.
3D Pythagoras and Trigonometry questions: Extended
Extended content, for Papers 2 and 4.
Question 2
A cuboid measures \(5\) cm by \(4\) cm by \(3\) cm.
Find the length of its longest diagonal, correct to 3 significant figures.
Show the worked answer
- \(d = \sqrt{25 + 16 + 9} = \sqrt{50} = 7.071\ldots\) cm
Answer: \(7.07\) cm
Question 3
A cube has edges of \(10\) cm.
Find the length of a diagonal joining opposite corners, correct to 3 significant figures.
Show the worked answer
- \(d = \sqrt{10^2 + 10^2 + 10^2} = \sqrt{300} = 17.32\ldots\) cm
Answer: \(17.3\) cm
Question 4
A cuboid has a base \(8\) cm by \(6\) cm and a height of \(5\) cm.
Find the angle between a longest diagonal and the base, correct to 1 decimal place.
Show the worked answer
- Diagonal of the base: \(\sqrt{8^2 + 6^2} = 10\) cm.
- \(\tan\theta = \frac{5}{10}\), so \(\theta = 26.56\ldots°\)
Answer: \(26.6°\)
Question 5
A cube has edges of \(6\) cm.
Find the angle between a diagonal joining opposite corners and the base, correct to 1 decimal place.
Show the worked answer
- Diagonal of the base: \(\sqrt{6^2 + 6^2} = 6\sqrt{2}\) cm.
- \(\tan\theta = \frac{6}{6\sqrt{2}} = \frac{1}{\sqrt{2}}\), so \(\theta = 35.26\ldots°\)
Answer: \(35.3°\)
Question 6
A pyramid has a square base of side \(8\) cm. Its apex is directly above the centre of the base, and each sloping edge is \(9\) cm long.
Find the height of the pyramid.
Show the worked answer
- Half the diagonal of the base: \(\frac{1}{2}\sqrt{8^2 + 8^2} = 4\sqrt{2}\) cm, and \((4\sqrt{2})^2 = 32\).
- Height \(= \sqrt{9^2 - 32} = \sqrt{49} = 7\) cm
Answer: \(7\) cm
Question 7
A pyramid has a square base of side \(8\) cm, its apex directly above the centre of the base and sloping edges of \(9\) cm.
Find the angle between a sloping edge and the base, correct to 1 decimal place.
Show the worked answer
- Height \(7\) cm and half-diagonal \(4\sqrt{2} = 5.656\ldots\) cm.
- \(\tan\theta = \frac{7}{5.656\ldots}\), so \(\theta = 51.05\ldots°\)
Answer: \(51.1°\)
Question 8
A rectangular field is \(30\) m by \(40\) m. A vertical pole \(12\) m tall stands at one corner.
Find the angle of elevation of the top of the pole from the opposite corner of the field, correct to 1 decimal place.
Show the worked answer
- Distance across the field between opposite corners: \(\sqrt{30^2 + 40^2} = 50\) m.
- \(\tan\theta = \frac{12}{50}\), so \(\theta = 13.49\ldots°\)
Answer: \(13.5°\)
Every question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).
Keep going
- Trigonometry practice for 0580: more questions, each one marked, tagged Core or Extended
- My mistakes, then a Skill Builder on the skills you missed
- Common mistakes in 0580: what the Cambridge examiner reports say
- Trigonometry revision notes and the 0580 formula sheet
- Cambridge IGCSE Maths 0580 past papers, when you are ready for a full paper
Questions students ask
Which Cambridge 0580 papers can test 3D Pythagoras and trigonometry?
3D Pythagoras and Trigonometry is Extended content (syllabus 6.6), so it is tested only on the Extended papers, Paper 2 (non-calculator) and Paper 4 (calculator). Core candidates do not need it.
Are these Cambridge past-paper questions?
No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.
Where can I practise more 3D Pythagoras and trigonometry questions?
In the 0580 practice for topic 6 (Trigonometry), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.