Question:
The base of a regular pyramid is a square and each of the other four sides is an equilateral triangle, length of each side being 20 cm. The vertical height of the pyramid, in cm, is
The base of a regular pyramid is a square and each of the other four sides is an equilateral triangle, length of each side being 20 cm. The vertical height of the pyramid, in cm, is
Updated On: Jul 28, 2025
- \(8\sqrt{3}\)
- 12
- \(5\sqrt{5}\)
- \(10\sqrt{2}\)
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The Correct Option is D
Solution and Explanation
From the diagram, it is evident that AB serves as both the height of the equilateral triangle and the slant height of the pyramid.
Given a regular triangle with side length 20 units:
- The length of the perpendicular from vertex \( A \) to base \( BC \) is: \[ AB = \frac{\sqrt{3}}{2} \times \text{side} = \frac{\sqrt{3}}{2} \times 20 = 10\sqrt{3} \]
- The median or half-diagonal from vertex \( A \) to centroid \( O \) is: \[ AO = \frac{1}{2} \times \text{side} = \frac{1}{2} \times 20 = 10 \]
Apply Pythagoras Theorem in triangle \( \triangle AOB \):
\[ OB^2 = AB^2 - AO^2 \] \[ OB^2 = (10\sqrt{3})^2 - 10^2 = 300 - 100 = 200 \] \[ OB = \sqrt{200} = 10\sqrt{2} \]
✅ Final Answer:
The height of the pyramid is: \[ \boxed{10\sqrt{2}} \]
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