Question 1: Calculate the length of the space diagonal of a cuboid with length 3 cm, width 4 cm, and height 12 cm.

Answer: 13

Question 2: A cuboid has dimensions 5 cm, 12 cm, and 13 cm. What is the length of its space diagonal?

Answer: 15

Question 3: A student claims that the diagonal of a cuboid with sides 6 cm, 8 cm, and 10 cm will always be less than 15 cm. Is this correct? Explain your reasoning.

Answer: Incorrect; the diagonal is 12.8 cm, which is less than 15 cm, so their claim is correct.

Question 4: Identify the mistake in the following calculation: 'Diagonal = √(6^2 + 8^2 + 10^2) = √(36 + 64 + 100) = √200 ≈ 14.14 cm'.

Answer: The mistake is taking the square root of the sum directly; the calculation is correct mathematically, but the student incorrectly assumes the Pythagoras' theorem for 3D. The correct approach is the same, but they need to understand it applies to space diagonals.

Question 5: A cuboid measures 8 cm by 15 cm by 17 cm. Calculate the length of the space diagonal.

Answer: 20

Question 6: A common misconception is that the diagonal of a cube with side length 6 cm is 6√2 cm. Identify and explain the error.

Answer: The error is confusing the face diagonal with the space diagonal. The face diagonal of a cube with side 6 cm is 6√2 cm, but the space diagonal is 6√3 cm.

Question 7: Construct a cuboid with dimensions 7 cm, 24 cm, and 25 cm. Calculate its space diagonal. Does this form a Pythagorean triple?

Answer: 26

Question 8: A rectangular box has a length of 9 cm, width of 40 cm, and height of 41 cm. If the student incorrectly computes the diagonal as √(9^2 + 40^2 + 41^2) = √(81 + 1600 + 1681) ≈ 58.33 cm, identify the misconception and correct it.

Answer: The misconception is treating the diagonal as the square root of the sum of squares of all three sides in a three-dimensional context. The correct diagonal length is √(9^2 + 40^2 + 41^2) = √(81 + 1600 + 1681) ≈ 58.33 cm. Actually, their calculation is correct; the misconception is misunderstanding the application of Pythagoras' theorem in 3D, which is valid here—so the mistake is assuming this calculation is incorrect.

Question 9: Given a cuboid with sides 10 cm, 24 cm, and 25 cm, determine whether the space diagonal forms a Pythagorean triple.

Answer: Yes, diagonal is 26 cm, which makes (10, 24, 26), a Pythagorean triple.

Question 10: Challenging: A cuboid has a space diagonal of 30 cm. Its length and width are 12 cm and 16 cm. Find the height and verify if the dimensions are consistent.

Answer: The height is √(30^2 - 12^2 - 16^2) = √(900 - 144 - 256) = √500 ≈ 22.36 cm.