The diagonals of three faces of a rectangular parallelepiped that meet at a single corner measure √5 cm, √10 cm, and √13 cm. What…

Question

The diagonals of three faces of a rectangular parallelepiped that meet at a single corner measure √5 cm, √10 cm, and √13 cm. What is the volume of this parallelepiped?

1. A. 18 cm³
2. B. 24 cm³
3. C. 12 cm³
4. D. 6 cm³ (Correct answer)

Correct Answer

Option D — 6 cm³

Detailed Solution & Explanation

The correct answer is 6 cm³.

Key Points

• Let the edges be a, b and c. The three face diagonals meeting at one corner give:
  • a² + b² = 5, b² + c² = 10, c² + a² = 13
• Adding all three: 2(a² + b² + c²) = 28, so a² + b² + c² = 14.
• Subtract each equation in turn:
  • c² = 14 − 5 = 9 → c = 3
  • a² = 14 − 10 = 4 → a = 2
  • b² = 14 − 13 = 1 → b = 1
• Volume = 2 × 1 × 3 = 6 cm³.

Additional Information

• Adding the three equations and halving is the standard route — attempting to solve them one at a time is far slower.
• The space diagonal of a cuboid is √(a² + b² + c²), here √14, a common follow-up question.
• Surface area would be 2(ab + bc + ca) = 2(2 + 3 + 6) = 22 cm².
• Note the numbers are chosen so that a² + b² + c² comes out whole and each edge is an integer — a sign you have read the diagonals correctly.