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.

Correct — A. Get the base area from the diagonal, then take one-third of base × height.

Square area from its diagonal: d² ÷ 2 = 512 ÷ 2 = 256 m²
Check: side = √512 ÷ √2 = √256 = 16 m, and 16² = 256

Volume = (1⁄3) × base area × height
= (1⁄3) × 256 × 8 = 2048⁄3
= 682.67 m³ ≈ 683 m³ → option (a)

Why the others are wrong

• (b)
  982 m³ — 982 m³ would need a base of 982 × 3 ÷ 8 ≈ 368 m². A square with diagonal √512 m has area 512 ÷ 2 = 256 m².
• (c)
  1092 m³ — 1092 m³ implies a base of 1092 × 3 ÷ 8 = 409.5 m², far above 256 m². Even dropping the 1⁄3 gives 2048 m³, not 1092.
• (d)
  782 m³ — 782 m³ implies a base of 782 × 3 ÷ 8 ≈ 293 m². The base is 16 m × 16 m = 256 m², which gives 682.67 m³.

A pyramid holds one-third of the prism with the same base and height: V = (1⁄3) × base area × height.

For a square given by its diagonal d, the side is d⁄√2, so the area is d²⁄2.

Here the diagonal is √512, so d² = 512 straight away and the area is 256 m² without finding the side.

Key facts

• Volume of a pyramid = (1⁄3) × base area × height.
• Area of a square from its diagonal = d²⁄2.
• √512 = 16√2, so the side is 16√2 ÷ √2 = 16 m.

Common traps

• Dropping the 1⁄3 and giving the prism volume, 256 × 8 = 2048 m³.
• Squaring the diagonal as if it were the side: 512 × 8 ÷ 3 ≈ 1365 m³.

No directly related past PYQ was found.