Pythagoras and Trigonometry in Three Dimensions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Finding the long diagonal of a cuboid

Mathematics · Geometry · ages 15-16
Name ______________________   Date ____________

  1. A cuboid is 9 cm by 12 cm and 20 cm tall. What is its long diagonal in cm?

    Answer: ______________

  2. A cuboid is 5 cm by 12 cm and 84 cm tall. What is its long diagonal?

    • 97 cm
    • 13 cm
    • 85 cm
    • 25 cm

  3. A cuboid is 3 cm by 4 cm by 12 cm. What is its long diagonal?

    • 5 cm
    • 13 cm
    • 12 cm

  4. Aisha says the long diagonal of a cuboid can be found by using Pythagoras twice, once on the base and once on the upright triangle. Is Aisha right?

    Circle one:   True   False

  5. A cuboid is 12 cm by 16 cm and 21 cm tall. What is its long diagonal in cm?

    Answer: ______________

  6. Sam has a right-angled triangle with hypotenuse 13 cm and one short side 5 cm. Sam drew no sketch and treated 13 as a short side. What should Sam have done?

    • Redrawn the triangle flat with the right angle marked
    • Added all three sides together, then halved the total
    • Measured the angle first with a protractor

  7. You need the long diagonal of a cuboid. What is the best first step?

    • Find the base diagonal
    • Add the three edge lengths together
    • Measure an angle
    • Find the volume

  8. Aisha says the space diagonal of a cuboid can be found by using Pythagoras twice, once on the base and once on the upright triangle. Is Aisha right?

    Circle one:   True   False

  9. A right square pyramid has base 12 cm and vertical height 8 cm, with the apex directly above the base centre. What is the angle between a lateral edge and the base, in degrees? Round your answer to 1 decimal place.

    • 36.9 degrees
    • 43.3 degrees
    • 53.1 degrees

  10. A cuboid is 24 cm by 32 cm and 42 cm tall. What is its long diagonal in cm?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Finding the long diagonal of a cuboid W1-mt_7yKxYqQHB2-s1

  1. 25 · Base 15 cm, then root of 225 plus 400 is 25.
  2. 85 cm · The base diagonal is 13 cm, and 13 squared plus 84 squared is 7225, whose root is 85.
  3. 13 cm · Base diagonal 5 cm first, then the upright triangle gives 13 cm.
  4. True · Once for the base diagonal, once for the upright triangle.
  5. 29 · Base 20 cm, then root of 400 plus 441 is 29.
  6. Redrawn the triangle flat with the right angle marked · A flat sketch with the hypotenuse labelled stops side mix ups.
  7. Find the base diagonal · The base diagonal plus the height forms the upright triangle that holds the long diagonal.
  8. True · Once for the base diagonal, once for the upright triangle.
  9. 43.3 degrees · The centre to corner is 6 root 2 cm, and tangent is 8 over 6 root 2, which gives about 43.3 degrees.
  10. 58 · Base 40 cm, then root of 1600 plus 1764 is 58.
Worksheet · LightMySky