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

We need to find the side length of a regular hexagon whose area equals the area of an equilateral triangle with side 12 cm.

We'll use the area formulas for both shapes and set them equal.

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For an equilateral triangle with side length $s$, the area formula is:

$\text{Area} = \tfrac{\sqrt{3}}{4} \times s^2$

An equilateral triangle can be split into two right triangles, and using the Pythagorean theorem, we can derive this formula.

With $s = 12$ cm:

$\begin{aligned} \text{Area of triangle} &= \tfrac{\sqrt{3}}{4} \times 12^2 \\ &= \tfrac{\sqrt{3}}{4} \times 144 \\ &= 36\sqrt{3} \end{aligned}$

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For a regular hexagon with side length $a$, the area formula is:

$\text{Area} = \tfrac{6\sqrt{3}}{4} \times a^2$

A regular hexagon can be divided into 6 equilateral triangles, each with side length $a$. Since each triangle has area $\tfrac{\sqrt{3}}{4} \times a^2$, the total area is:

$6 \times \tfrac{\sqrt{3}}{4} \times a^2 = \tfrac{6\sqrt{3}}{4} \times a^2$

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Since the areas are equal:

$\tfrac{6\sqrt{3}}{4} \times a^2 = \tfrac{\sqrt{3}}{4} \times 144$

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Both sides by 4:

$6\sqrt{3} \times a^2 = \sqrt{3} \times 144$

Both sides by $\sqrt{3}$:

$6a^2 = 144$

Both sides by 6:

$a^2 = \tfrac{144}{6} = 24$

$a = \sqrt{24}$

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To simplify $\sqrt{24}$:

$\begin{aligned} \sqrt{24} &= \sqrt{4 \times 6} \\ &= \sqrt{4} \times \sqrt{6} \\ &= 2\sqrt{6} \end{aligned}$

Therefore, the side length of the hexagon is $2\sqrt{6}$ cm.