The height of a pyramid is 8 m, and the base of the pyramid is a square whose diagonal is √512 m. Find the approximate volume of…

Question

The height of a pyramid is 8 m, and the base of the pyramid is a square whose diagonal is √512 m. Find the approximate volume of the pyramid.

1. A. 683 m³ (Correct answer)
2. B. 1092 m³
3. C. 982 m³
4. D. 782 m³

Correct Answer

Option A — 683 m³

Detailed Solution & Explanation

The correct answer is 683 m³.

Key Points

• Get the base area from the diagonal. For a square, area = d²/2:
  • area = (√512)² / 2 = 512 / 2 = 256 m²
• Apply the pyramid formula V = (1/3) × base area × height:
  • V = (1/3) × 256 × 8 = 2048/3 = 682.67 ≈ 683 m³

Additional Information

• The d²/2 shortcut avoids finding the side at all. It follows from d = a√2, so a² = d²/2.
• A pyramid is exactly one-third of the prism sharing its base and height — the same relationship a cone has to its cylinder.
• Related formulae: lateral surface area = ½ × perimeter of base × slant height, and total surface area adds the base.
• The slant height l, the vertical height h and the apothem of the base form a right triangle, so l² = h² + (a/2)² for a square pyramid.
• Note the trap: using the diagonal as though it were the side gives 512 × 8/3, which is nowhere near any option.