CAT 2019

Slot 2

QA

Question & Solution

1. Concept Used

• Topic: 3D Geometry — Pyramid with Square Base and Equilateral Triangle Faces
• Formula: $$ h^2 + r^2 = l^2 $$ where $h$ = vertical height of pyramid, $r$ = distance from centre of base to midpoint of a base edge (apothem of base), and $l$ = slant height (height of equilateral triangular face).

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2. Calculation

Each side of the pyramid (both the square base and the four equilateral triangular faces) has length $20$ cm.

Since the base is a square of side $20$ cm, the centre of the base (apex projected point) is equidistant from all four sides. The distance from the centre of the square to the midpoint of any side is half the side length: $$r = \frac{20}{2} = 10 \text{ cm}$$

The slant height $l$ of the pyramid is the height of one of the equilateral triangular faces. For an equilateral triangle of side $20$ cm, the height is: $$l = \frac{\sqrt{3}}{2} \times 20 = 10\sqrt{3} \text{ cm}$$

Now, the vertical height $h$, the apothem $r$, and the slant height $l$ form a right triangle (by the Pythagorean theorem): $$h^2 + r^2 = l^2$$ $$h^2 + (10)^2 = (10\sqrt{3})^2$$ $$h^2 + 100 = 300$$ $$h^2 = 200$$ $$h = \sqrt{200} = 10\sqrt{2} \text{ cm}$$

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3. Solution

Answer = Option B ✅

The vertical height of the pyramid is $10\sqrt{2}$ cm.