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      Applying Pythagoras' theorem in 3D

      Lesson details

      Learning outcome

      I can apply knowledge of Pythagoras' theorem to 3D problems.

      Key learning points

      1. For packing efficiency, it can be useful to know the length of the longest diagonal
      2. This would be from a bottom corner of a rectangular box, to the opposite top corner
      3. You can construct a right-angled triangle to calculate this
      4. You will need to know the length of the diagonal of the base of the box
      5. There is a formula you can deduce to calculate the longest diagonal

      Keywords

      • Hypotenuse - The hypotenuse is the side of a right-angled triangle which is opposite the right angle.

      • Pythagoras' theorem - Pythagoras’ theorem states that the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the hypotenuse.

      • Cross-section - A cross-section is a 2D face made from cutting straight through any plane of a 3D object.

      Common misconception

      Pythagoras' theorem cannot be applied to 3D shapes.

      Pupils may benefit from imaging the cuboid is 'sliced' through and a triangle drawn on the now visible face.

      Teacher tip

      Pupils may benefit from using the frame of a cuboid and some string so they can physically construct the right-angled triangles within it.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      The relationship between the hypotenuse and the two short sides on a right-angled triangle is known as ' theorem.

      Correct Answer: Pythagoras

      Q2.
      Work out the the length of the hypotenuse, to 1 decimal place.

      An image in a quiz
      Correct Answer: 8.1 cm, 8.1

      Q3.
      Work out the the length of the hypotenuse, to 1 decimal place.

      An image in a quiz
      Correct Answer: 43.1 cm, 43.1

      Q4.
      Work out the length of the third side of the triangle, to 3 significant figures.

      An image in a quiz
      Correct Answer: 25.3 cm, 25.3

      Q5.
      Work out the length of the third side of the triangle, to 3 significant figures.

      An image in a quiz
      Correct Answer: 5.99 cm, 5.99

      Q6.
      A football pitch is 115 yards by 74 yards. What is the distance diagonally from one corner flag to the other? Give your answer to the nearest yard.

      Correct Answer: 137, 137 yards, 137 yd

      6 Questions

      Q1.
      Which of these are right-angled triangles?

      An image in a quiz
      Correct Answer: An image in a quiz
      An image in a quiz
      Correct Answer: An image in a quiz
      An image in a quiz

      Q2.
      Given that DE = 6 cm and EH = 15 cm, what is the length of DH, to 1 decimal place?

      An image in a quiz
      Correct Answer: 16.2 cm, 16.2

      Q3.
      Given that BG = 9 cm and GE = $$7\sqrt{2}$$ cm, what is the length of BE, to 1 decimal place?

      An image in a quiz
      Correct Answer: 13.4 cm, 13.4

      Q4.
      If AF, FE and FH are known, which other length do we need, in order to calculate the volume of this cuboid?

      An image in a quiz
      BG
      BC
      BE
      Correct answer: EH
      AG

      Q5.
      Given that AF = 5 cm, FE = 12 cm and FH = 15 cm, work out the length of EH.

      An image in a quiz
      Correct Answer: 9 cm, 9, nine

      Q6.
      Given that AF = 5 cm, FE = 12 cm and FH = 15 cm, the volume of the cuboid ABCDEFGH is $$\text{ cm}^3$$

      An image in a quiz
      Correct Answer: 540

      To help you plan your 10 maths lesson on: Applying Pythagoras' theorem in 3D, download all teaching resources for free and adapt to suit your pupils' needs...