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
\(4 \sqrt6\)
\(6 \sqrt6\)
\(2 \sqrt6\)
\(\sqrt6\)
The Correct Option is C
Solution and Explanation
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.
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