Question

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

• A. \(4 \sqrt6\)
• B. \(6 \sqrt6\)
• C. \(2 \sqrt6\)
• D. \(\sqrt6\)

Answer 1

The Correct Option is C

Let's break this down: 

The area of an equilateral triangle with side 

\( a \) is given by: \[ \text{Area of triangle} = \frac{\sqrt{3}}{4} a^2 \] 

For the given equilateral triangle of side 12 cm, the area is: 

\[ \text{Area} = \frac{\sqrt{3}}{4} (12^2) = 36\sqrt{3} \] sq.cm 

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

So, the area of one of these equilateral triangles with side \( s \) is: \[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} s^2 \] 

The area of the hexagon, which is the sum of the areas of the 6 equilateral triangles, is: 

\[ \text{Area of hexagon} = 6 \times \frac{\sqrt{3}}{4} s^2 = \frac{3\sqrt{3}}{2} s^2 \] 

Given that the area of the hexagon is equal to the area of the equilateral triangle of side 12 cm: 

\[ \frac{3\sqrt{3}}{2} s^2 = 36\sqrt{3} \] 

[ s^2 = 24 \] 

\[ s = 2\sqrt{6} \] 

So, the length of each side of the hexagon is: 2√6.