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Trigonometry in Three Dimensions

Calculate sides and angles in three dimensional shapes using trigonometry (including Pythagoras' theorem).

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This is level 4: Worded three-dimensional problems without diagrams. You can earn a trophy if you get at least 4 questions correct and you do this activity online. Give answers correct to three significant figures.

1. Find the angle that the diagonal from a vertex of a cube makes with an edge of the cube at that vertex.

° Correct Wrong

2. The pyramid ABCDE has a square base ABCD with sides of length 43m and AE = AB = AC = AD = 63m. Find the height of the pyramid.

m Correct Wrong

3. A triangular prism has a cross section in the shape of a right-angled triangle with sides of length 36cm, 48cm and 60cm. The length of the prism is 32cm. PQ is the line from the vertex with the smallest angle of one of the triangular faces to the the vertex with the right angle of the other triangular face. RQ is the edge joining the two vertices at the right angles of the triangular faces. Find the angle between PQ and RQ.

° Correct Wrong

4. A large sports hall is built in the shape of a massive cuboid with a height of 44m, width 30m and length 58m. Find the angle of elevation of one of the top corners of the sports hall as seen from the diagonally opposite corner on the floor of the building.

° Correct Wrong

5. The angle of elevation of the top of a building is 15 degrees from a point X which is due north of the building. The angle of elevation of the top of that same building is 33 degrees from a point Y which is due west of the building. Find the height of the building in metres if the point X is one kilometre further from the building than the point Y.

m Correct Wrong

This is Trigonometry in Three Dimensions level 4. You can also try:
Level 1 Level 2 Level 3 Level 5


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Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

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Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.


If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

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Description of Levels



Pythagoras' Theorem - Start by applying Pythagoras' theorem in two dimensions.

2D Trigonometry - Revise using trigonometry in two dimensions.

Level 1 - Three dimensional problems which can be solved using Pythagoras' theorem

Level 2 - More three dimensional problems requiring the use of trigonometrical ratios

Level 3 - Mixed three dimensional problems

Level 4 - Worded three-dimensional problems without diagrams

Level 5 - Problems involving planes (not the flying type!)

Exam-Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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You may also want to use a calculator to check your working. See Calculator Workout skill 11.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Log in Sign up