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

• A. \(8\sqrt{3}\)
• B. 12
• C. \(5\sqrt{5}\)
• D. \(10\sqrt{2}\)

Answer 1

The Correct Option is D

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: \[ \boxed{10\sqrt{2}} \]