Question:medium

If the area of a regular hexagon is equal to the area of an equilateral triangle of side 12 cm, then the length, in cm, of each side of the hexagon is

Updated On: Jan 15, 2026

  • \(4 \sqrt6\)

  • \(6 \sqrt6\)

  • \(2 \sqrt6\)

  • \(\sqrt6\)

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The Correct Option is C

Solution and Explanation

Analysis:

The area of an equilateral triangle with side length \( a \) is calculated using the formula: \[ \text{Area of triangle} = \frac{\sqrt{3}}{4} a^2 \]

For an equilateral triangle with a side length of 12 cm, the area is: \[ \text{Area} = \frac{\sqrt{3}}{4} (12^2) = 36\sqrt{3} \] square centimeters. 

A regular hexagon with side length \( s \) can be divided into 6 equilateral triangles, each with a side length of \( s \). 

The area of one such equilateral triangle is: \[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} s^2 \] 

The total area of the hexagon, being the sum of the areas of these 6 triangles, is: \[ \text{Area of hexagon} = 6 \times \frac{\sqrt{3}}{4} s^2 = \frac{3\sqrt{3}}{2} s^2 \] 

Given that the hexagon's area equals the area of the equilateral triangle with a side of 12 cm: \[ \frac{3\sqrt{3}}{2} s^2 = 36\sqrt{3} \] 

Solving for \( s^2 \): \[ s^2 = 24 \] 

Solving for \( s \): \[ s = 2\sqrt{6} \] 

Conclusion: The side length of the hexagon is 2√6.

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