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

Answer: (C)

$2 \sqrt6$

Answer 2

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.