Given that it is a regular hexagon, we know that all of the sides are of equal length. Therefore, all 6 of the triangles (we get from drawing lines to opposite vertexes) are congruent triangles.

Lets use the fact that there are 360 degrees in a full rotation. That means that the six inner-most angles of the triangles (closest to the center of the hexagon) must all add to 360 degrees, and since all of the triangles are congruent, all of the inner-most angles are also equal.

\(\displaystyle \frac{360}{6}=60^{\circ}\)

If the innermost angle is 60 degrees, and the fact that it is a regular hexagon, we can therefore state that the other two angles are

\(\displaystyle \frac{\left ( 180-60 \right )}{2}=60^{\circ}\) 

as well. Since all of the angles are 60 degrees, it is an equilateral triangle.

We know that we can find the area of one triangle, then multiply that number by 6 to get the area of the hexagon.

With that in mind, we look at the right triangle.

Look familiar?? The 30-60-90 and 45-45-90 right triangles appear often, so it's worth it to remember the shortcuts to the Pythagorean Theorem so you don't have to do the calculations every time.

For our numbers we have  \(\displaystyle \mathbf{a}\sqrt{3} = 7\sqrt{3}}\), so our base is going to be 7 cm.

The base of the larger equilateral triangle is going to be twice as big, so

\(\displaystyle 7\cdot 2=14\ cm\).

Plug these into

\(\displaystyle A=base\cdot height\cdot \frac{1}{2}\)

and then multiply by 6 (remember we want the area of all 6 triangles inside of the hexagon)

\(\displaystyle 14 \cdot 7\sqrt{3} \cdot \frac{1}{2} = 49\sqrt{3}}\)

\(\displaystyle 49\sqrt{3} \cdot 6 = 294\sqrt{3}\approx 509.22\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(4^2)=\frac{3\sqrt3}{2}(16)=\frac{48\sqrt3}{2}=24\sqrt3\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(5^2)=\frac{3\sqrt3}{2}(25)=\frac{75\sqrt3}{2}\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(2^2)=\frac{3\sqrt3}{2}(4)=\frac{12\sqrt3}{2}=6\sqrt3\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(6^2)=\frac{3\sqrt3}{2}(36)=\frac{108\sqrt3}{2}=54\sqrt3\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(10^2)=\frac{3\sqrt3}{2}(100)=\frac{300\sqrt3}{2}=150\sqrt3\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(7^2)=\frac{3\sqrt3}{2}(49)=\frac{147\sqrt3}{2}\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(2x)^2=\frac{3\sqrt3}{2}(4x^2)=\frac{12x^2\sqrt3}{2}=6x^2\sqrt3\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(12x)^2=\frac{3\sqrt3}{2}(144x^2)=\frac{432x^2\sqrt3}{2}=216x^2\sqrt3\)

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}side^2\)

Now, substitute in the value for the side length.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}(9x)^2=\frac{3\sqrt3}{2}(81x^2)=\frac{243x^2\sqrt3}{2}\)