Use the formula for perimeter to solve for the length of a side of the regular hexagon:

\(\displaystyle P=6s\)

Where \(\displaystyle P\) is perimeter and \(\displaystyle s\) is the length of a side.

In this case:

\(\displaystyle 112.8=6s\)

\(\displaystyle s=\frac{112.8}{6}=18.8\)

Use the formula for perimeter to solve for the side length:

\(\displaystyle P=6s\)

\(\displaystyle 85.8=6s\)

\(\displaystyle s=14.3\)

Use the formula for perimeter to solve for the side length:

\(\displaystyle P=6s\)

\(\displaystyle 114=6s\)

\(\displaystyle s=19\)

Use the formula for perimeter to solve for the side length:

\(\displaystyle P=6s\)

\(\displaystyle 54=6s\)

\(\displaystyle s=9\)

To find the side of a hexagon given the area, set the area formula equal to the given area and solve for the side.
\(\displaystyle 216\sqrt{3}=\frac{3}{2}\sqrt{3}*a^2 \newline \newline 432\sqrt3 = 3\sqrt{3}*a^2 \newline \newline 432 = 3*a^2 \newline 144 = a^2 \newline 12 = a\)

In a regular hexagon, all of the sides are the same length, and all of the angles are equivalent. The problem tells us that all of the angles inside the hexagon sum to \(\displaystyle 720^{\circ}\). To find the value of one angle, we must divide \(\displaystyle 720^{\circ}\) by \(\displaystyle 6\), since there are \(\displaystyle 6\) angles inside of a hexagon. 

\(\displaystyle \frac{720 ^{\circ}}{6}=120^{\circ}\)

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

\(\displaystyle 180*(s-2)\), where \(\displaystyle s\) is the number of sides.

Therefore, a hexagon like this one has:

\(\displaystyle 180*4=720^{\circ}\).

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles. This lets you draw the following figure:

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to \(\displaystyle 720\). Since there are \(\displaystyle 6\) angles, you know that the numeric portion will be \(\displaystyle 180*6\) or \(\displaystyle 1080\). Thus, you can write:

\(\displaystyle 1080-a-2a-a-3a-a-3a=720\)

Simplify and solve for \(\displaystyle a\):

\(\displaystyle 1080-11a=720\)

\(\displaystyle -11a=-360\)

\(\displaystyle a=\frac{360}{11}\)

This is \(\displaystyle 32.72727272727272...\) or \(\displaystyle 32.73^{\circ}\).

There are two key things for a question like this. The first is to know that a polygon has a total degree measure of:

\(\displaystyle 180*(s-2)\), where \(\displaystyle s\) is the number of sides.

Therefore, a hexagon like this one has:

\(\displaystyle 180*4=720^{\circ}\).  

Next, you should remember that all of the exterior angles listed are supplementary to their correlative interior angles.  This lets you draw the following figure: 

Now, you just have to manage your algebra well. You must sum up all of the interior angles and set them equal to \(\displaystyle 720\). Thus, you can write:

\(\displaystyle 140+160+150+85+135+x=720\)

Solve for \(\displaystyle x\):

\(\displaystyle 670+x=720\)

\(\displaystyle x=50^{\circ}\)

First, let's draw out the hexagon.

Because the hexagon is made up of 6 equilateral triangles, to find the area of the hexagon, we will first find the area of each equilateral triangle then multiply it by 6.

Using the Pythagorean Theorem, we find that the height of each equilateral triangle is \(\displaystyle 4\sqrt3\).

The area of the triangle is then

\(\displaystyle \frac{8 \times 4\sqrt3}{2}=16\sqrt3\)

Multiply this value by 6 to find the area of the hexagon.

This question is asking about the area of a regular hexagon that looks like this:

Now, you could proceed by noticing that the hexagon can be divided into little equilateral triangles:

By use of the properties of isosceles and \(\displaystyle 30-60-90\) triangles, you could compute that the area of one of these little triangles is:

\(\displaystyle \frac{\sqrt{3}}{4}s^2\), where \(\displaystyle s\) is the side length. Since there are \(\displaystyle 6\) of these triangles, you can multiply this by \(\displaystyle 6\) to get the area of the regular hexagon:

\(\displaystyle 6*\frac{\sqrt{3}}{4}s^2=\frac{3\sqrt{3}}{2}s^2\)

It is likely easiest merely to memorize the aforementioned equation for the area of an equilateral triangle. From this, you can derive the hexagon area equation mentioned above. Using this equation and our data, we know:

\(\displaystyle Area=\frac{3\sqrt{3}}{2}*6^2=\frac{3\sqrt{3}}{2}*36=54\sqrt{3}\)