Find the area of a regular hexagon whose side measures 14√3 cm.

The correct answer is 882√3 cm².

To find the area of a regular hexagon, we can break it down into smaller, simpler geometric shapes. A regular hexagon is made up of 6 identical (congruent) equilateral triangles.

First, let's write down the formula for the area of an equilateral triangle with side length $s$:

$\text{Area of an equilateral triangle}=\frac{\sqrt{3}}{4}{s}^2$

Since a regular hexagon consists of 6 such equilateral triangles, the total area of the hexagon is given by:

$\text{Area of hexagon}=6\times \left(\frac{\sqrt{3}}{4}{s}^2\right)=\frac{3\sqrt{3}}{2}{s}^2$

Given in the problem, the side length of the regular hexagon is:

$s=14\sqrt{3}\text{cm}$

Now, let's substitute the value of $s$ into the area formula step-by-step.

Step 1: Calculate the square of the side length ($s^2$):

$s^2={\left(14\sqrt{3}\right)}^2={14}^2\times {\left(\sqrt{3}\right)}^2$

$s^2=196\times 3=588$

Step 2: Substitute $s^2=588$ into the hexagon area formula:

$\text{Area}=\frac{3\sqrt{3}}{2}\times 588$

Step 3: Divide 588 by 2:

$\frac{588}{2}=294$

Step 4: Multiply 294 by $3\sqrt{3}$:

$\text{Area}=3\times 294\times \sqrt{3}=882\sqrt{3}{\text{cm}}^2$

Therefore, the area of the regular hexagon is 882√3 cm².