Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above. The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is

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GATE PI 2021 Official Paper

Option 1 : 2 : 3

__Concept:__

Area of an equilateral triangle = \(\frac{\sqrt3}{4}\ \times\ a^2\)

where a = side of the triangle.

__Calculation:__

__Given:__

Let the side of the large triangle is **'a' **then the side of the regular hexagon is \(\frac{a}{3}\)

If each triangle's area = \(\frac{\sqrt3}{4}\ \times\ a^2\) then the area of regular hexagon = \(6\ \times\ \frac{\sqrt{3}}{4}\ \times\ \left ( \frac{a}{3} \right )^2\)

Required ratio = \(\frac{Area \ of\ regular\ hexagon}{Area\ of\ equilateral \ triangle}\) = \(\frac{A_H}{A_T}\)

⇒ \(\frac{A_H}{A_T}\ =\ \frac{6\ \times\ \frac{\sqrt{3}}{4}\ \times\ \left ( \frac{a}{3} \right )^2}{\frac{\sqrt3}{4}\ \times\ a^2}\ =\ \frac{6}{9}\ =\ \frac{2}{3}\)

**∴ **The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is \(\frac{2}{3}\)