The base of a right pyramid is an equilateral triangle with side 10 cm, and its vertical height is 24 cm. Find the volume (in cm³…

Question

The base of a right pyramid is an equilateral triangle with side 10 cm, and its vertical height is 24 cm. Find the volume (in cm³) of the pyramid.

1. A. 200√3 cm³ (Correct answer)
2. B. 100√3 cm³
3. C. 400√3 cm³
4. D. 150√3 cm³

Correct Answer

Option A — 200√3 cm³

Detailed Solution & Explanation

The correct answer is 200√3 cm³.

Key Points

• The base is an equilateral triangle of side 10 cm, so its area is:
  • (√3/4) × a² = (√3/4) × 100 = 25√3 cm²
• Apply V = (1/3) × base area × height:
  • V = (1/3) × 25√3 × 24 = 25√3 × 8 = 200√3 cm³

Additional Information

• The equilateral triangle formulae worth automatic recall: area = (√3/4)a², height = (√3/2)a, inradius = a/(2√3) and circumradius = a/√3.
• A pyramid is one-third of the prism on the same base and of the same height, whatever the base shape.
• Since the height here is a multiple of 3, the division is exact and the surd survives into the answer — which is why every option carries √3.
• For a right pyramid the apex sits directly above the centre of the base, so the vertical height is used directly; an oblique pyramid would need the perpendicular distance instead.