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3D Pythagoras & trigonometry practice questions with answers

Higher · Grade 7 · Calculator

A second set of 3D Pythagoras & trigonometry questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

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Question 1 Easy

A cuboid has base 6 by 8 and height 4. Find the tan of the angle between the space diagonal and the base.

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Answer: 2/5

  1. The base diagonal is 10.
  2. tan of the angle is 4/10 = 2/5.
  • Method (1): The base diagonal is 10.
  • Accuracy (1): Answer 2/5

Question 2 Easy

A cuboid has base 3 by 4 and height 12. Find the sin of the angle between the space diagonal and the base.

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Answer: 12/13

  1. The base diagonal is 5 and the space diagonal is 13.
  2. sin of the angle is 12/13.
  • Method (1): The base diagonal is 5 and the space diagonal is 13.
  • Accuracy (1): Answer 12/13

Question 3 Medium

The face diagonal is 10 and one edge of that face is 6. Find the other edge.

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Answer: 8

  1. 10² − 6² = 64.
  2. The other edge is 8.
  • Method (1): 10² − 6² = 64.
  • Accuracy (1): Answer 8

Question 4 Medium

The base diagonal is 8 and the angle with the base is 30 degrees. Find the height to 3 significant figures.

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Answer: 4.62

  1. Height = 8 tan 30.
  2. tan 30 = 1/√3.
  • Method (1): Height = 8 tan 30.
  • Accuracy (1): Answer 4.62

Question 5 Hard

The space diagonal is 11 and the base is 6 by 6. Find the height.

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Answer: 7

  1. 6² + 6² = 72.
  2. 11² − 72 = 49, so the height is 7.
  • Method (1): 6² + 6² = 72.
  • Accuracy (1): Answer 7

Question 6 Hard

A cuboid is 6 by 8 by 5. Find the angle between the space diagonal and the base to the nearest degree.

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Answer: 27

  1. The base diagonal is 10.
  2. The angle is about 27 degrees.
  • Method (1): The base diagonal is 10.
  • Accuracy (1): Answer 27

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 13 marks

The face diagonal is 13 and one edge of that face is 5. Find the other edge.

Show answer

Answer: 12

  1. 13² − 5² = 144.
  2. The other edge is 12.
  • Method (1): 13² − 5² = 144.
  • Method (1): The other edge is 12.
  • Accuracy (1): Answer 12

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

What is the face diagonal of a 3 by 4 rectangle?

  1. 5
  2. 7
  3. 12
  4. √7
Show answer

5 √(3² + 4²) = √(9 + 16) = √25 = 5.

Check 2

What is the space diagonal of a 3 by 4 by 12 cuboid?

  1. 13
  2. 5
  3. 12
  4. 19
Show answer

13 √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13.

Check 3

Why is the space diagonal of a 3 by 4 by 12 cuboid 13 and not 5?

  1. 5 is only the base diagonal; the height 12 must be included
  2. the sides must be multiplied
  3. 5 is the height
  4. 13 is the perimeter
Show answer

5 is only the base diagonal; the height 12 must be included 5 is the 2D diagonal of the base; 3D diagonal incorporates the height 12.

Check 4

If all dimensions of a 3 by 4 by 12 cuboid are doubled to 6 by 8 by 24, what is the space diagonal?

  1. 26
  2. 13
  3. 52
  4. 24
Show answer

26 Doubling each dimension doubles the space diagonal: 2 × 13 = 26.

Check 5

In a cuboid with height 12 and base diagonal 5, what is tan of the angle with the base?

  1. 12/5
  2. 5/12
  3. 12/13
  4. 5/13
Show answer

12/5 tan θ = opposite / adjacent = height / base diagonal = 12/5.

Check 6

In that same cuboid (height 12, space diagonal 13), what is sin of the angle with the base?

  1. 12/13
  2. 5/13
  3. 12/5
  4. 13/12
Show answer

12/13 sin θ = opposite / hypotenuse = height / space diagonal = 12/13.

Check 7

Which line segment acts as the adjacent side for the angle between space diagonal and base?

  1. the base diagonal
  2. any base edge
  3. the height
  4. the space diagonal
Show answer

the base diagonal The angle lies between the space diagonal and the base plane, so base diagonal is adjacent.

Check 8

What is the diagonal of a 6 by 8 rectangle?

  1. 10
  2. 14
  3. 48
  4. √14
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10 √(6² + 8²) = √(36 + 64) = √100 = 10.

Check 9

What is the diagonal of a 5 by 12 rectangle?

  1. 13
  2. 17
  3. 60
  4. √17
Show answer

13 √(5² + 12²) = √(25 + 144) = √169 = 13.

Check 10

What is the space diagonal of a cuboid with edges 9, 12 and 20?

  1. 25
  2. 41
  3. 300
  4. 21
Show answer

25 √(81 + 144 + 400) = √625 = 25.

Check 11

A student calculates tan θ as 12/13 instead of 12/5. Which ratio did they calculate?

  1. sin θ (height / space diagonal)
  2. cos θ (base / space diagonal)
  3. tan θ upside down
  4. perimeter ratio
Show answer

sin θ (height / space diagonal) 12/13 is opposite / hypotenuse, which is sine.

Check 12

What happens to the space diagonal of a cuboid when every edge is multiplied by k?

  1. it is multiplied by k
  2. it is multiplied by k²
  3. it remains unchanged
  4. it is multiplied by √k
Show answer

it is multiplied by k Lengths scale linearly by the scale factor k.

Check 13

Work out √(81 + 144). What length does this represent for a 9 by 12 rectangle?

  1. 15 (face diagonal)
  2. 21 (perimeter)
  3. 108 (area)
  4. 13 (space diagonal)
Show answer

15 (face diagonal) √(81 + 144) = √225 = 15.

Check 14

In a cuboid with base 6 by 8 and height 24, what is sin of the angle with the base?

  1. 12/13
  2. 5/13
  3. 12/5
  4. 24/25
Show answer

12/13 Base diagonal = 10, space diagonal = 26. sin θ = 24/26 = 12/13.

Check 15

In a cuboid with base 6 by 8 and height 24, what is tan of the angle with the base? Give it as a fraction in its simplest form.

  1. 12/5
  2. 5/12
  3. 12/13
  4. 24/10
Show answer

12/5 tan θ = 24/10 = 12/5.

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