The perimeter in this question is irrelevant. Use the interior angle formula to determine the total sum of the angles in a hexagon.

\(\displaystyle \Theta =(n-2)\cdot180\)

There are six interior angles in a hexagon.

\(\displaystyle \Theta =(6-2)\cdot180 = 720^{\circ}\)

Each angle will be a sixth of the total angle.

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

Therefore, the sum of three angles in a hexagon is:

\(\displaystyle 120^{\circ}+120^{\circ}+120^{\circ}=360^{\circ}\)

Use the interior angle formula to find the total sum of angles in a pentagon.

\(\displaystyle \sum \Theta=(n-2) \cdot 180\)

\(\displaystyle n=5\) for a pentagon, so substitute this value into the equation and solve:

\(\displaystyle \sum \Theta=(5-2) \cdot 180 = 540^{\circ}\)

Divide this number by 5, since there are five interior angles.

\(\displaystyle \frac{540^{\circ}}{5}=108^{\circ}\)

The sum of four interior angles in a regular pentagon is:

\(\displaystyle 108^{\circ}+108^{\circ}+108^{\circ}+108^{\circ} =432^{\circ}\)

The perimeter of a regular pentagon has no effect on the interior angles of the pentagon.

Use the following formula to solve for the sum of all interior angles in the pentagon.

\(\displaystyle \sum\theta=(n-2)\cdot 180\)

Since there are 5 sides in a pentagon, substitute the side length \(\displaystyle n=5\).

\(\displaystyle \sum\theta=(5-2)\cdot 180=540\)

Divide this by 5 to determine the value of each angle, and then multiply by 2 to determine the sum of 2 interior angles.

\(\displaystyle \frac{540}{5}=108\)

\(\displaystyle 108*2=216\)

The sum of 2 interior angles of a pentagon is \(\displaystyle 216\).

The pentagon has 5 sides. To find the value of the interior angle of a pentagon, use the following formula to find the sum of all interior angles.

\(\displaystyle \sum\theta=(n-2)\cdot 180\)

Substitute \(\displaystyle n=5\).

\(\displaystyle \sum\theta=(5-2)\cdot 180=540\)

Divide this number by 5 to determine the value of each interior angle.

\(\displaystyle \frac{540}{5}=108\)

Every interior angle is 108 degrees.  The problem states that an interior angle is \(\displaystyle 12x\).  Set these two values equal to each other and solve for \(\displaystyle x\).

\(\displaystyle 12x=108\)

\(\displaystyle x=9\)

Area has no effect on the value of the interior angles of a pentagon. To find the sum of all angles of a pentagon, use the following formula, where \(\displaystyle n\) is the number of sides:

\(\displaystyle \sum\theta=(n-2)\cdot 180\)

There are 5 sides in a pentagon.  

\(\displaystyle \sum\theta=(5-2)\cdot 180=540\)

Divide this number by 5 to determine the value of each angle.

\(\displaystyle \frac{540}{5}=108^\circ\)

A regular polygon with \(\displaystyle N\) sides has \(\displaystyle N\) congruent angles, each of which measures 

\(\displaystyle \frac{(N-2)180^{\circ }}{N}\)

Setting \(\displaystyle N = 5\), the common angle measure can be calculated to be

\(\displaystyle \frac{(5-2)180^{\circ }}{5} = \frac{3 \cdot 180^{\circ }}{5} = \frac{540^{\circ }}{5} = 108^{\circ }\)

The statement is therefore false.

If one exterior angle is taken at each vertex of any polygon, and their measures are added, the sum is \(\displaystyle 360^{\circ }\). Each exterior angle of a regular pentagon has the same measure, so if we let \(\displaystyle t\) be that common measure, then

\(\displaystyle 5t = 360 ^{\circ }\)

Solve for \(\displaystyle t\):

\(\displaystyle 5t \div 5 = 360 ^{\circ } \div 5\)

\(\displaystyle t = 72 ^{\circ }\)

The statement is true.

Suppose Pentagon \(\displaystyle PENTA\) is regular. Each angle of a regular polygon of \(\displaystyle N\) sides has measure

\(\displaystyle \frac{(N-2)180^{\circ }}{N}\)

A pentagon has 5 sides, so set \(\displaystyle N = 5\); each angle of the regular hexagon has measure  

\(\displaystyle \frac{(5-2)180^{\circ }}{5} = \frac{ 3 \cdot 180^{\circ }}{5} = 108^{\circ }\)

Since one angle is given to be of measure \(\displaystyle 108 ^{\circ }\), the pentagon might be regular - but without knowing more, it cannot be determined for certain. Therefore, the correct choice is "undetermined".

From the given information, we can imagine the following image

In order to solve for the perimeter, we need to solve for the length of one of the sides. This can be accomplished by creating a right triangle from the apothem and the top angle that's marked. 

This angle measure can be calculated through:

\(\displaystyle \frac{360^{\circ}}{5} = 72^{\circ}\), where the sum of all the angles around the center of the pentagon sum up to \(\displaystyle 360^{\circ}\) and the \(\displaystyle 5\) is for the number of sides. We can do this because all of the interior angles of the pentagon will be equal because it is regular. Then, this answer needs to be divided by \(\displaystyle 2\) because the pictured right triangle is half of a larger isosceles triangle. 

\(\displaystyle \frac{72^{\circ}}{2}=36^{\circ}\)

Therefore, the marked angle is \(\displaystyle 36^{\circ}\). 

Now that one side and one angle are known, we can use SOH CAH TOA because this is a right triangle. In this case, the tangent function will be used because the mystery side of interest is opposite of the known angle, and we are already given the adjacent side (the apothem). In solving for the unknown side, we will label it as \(\displaystyle x\).

\(\displaystyle \tan(\theta )=\frac{opp}{adj}\)

\(\displaystyle \tan (36^{\circ})=\frac{x}{3}\)

\(\displaystyle 3 \cdot \tan (36^{\circ})=x\)

\(\displaystyle x=2.17963\)

For a more precise answer, keep the entire number in your calculator. This will prevent rounding errors. 

Keep in mind that the base of the triangle is actually only half the length of the side. This means that the value for \(\displaystyle x\) must be multiplied by \(\displaystyle 2\). 

\(\displaystyle 2.17963...\cdot 2 = 4.3596...\)

In order to solve for the perimeter, the length of the side needs to be multiplied by the number of sides; in this case, there are five sides.

\(\displaystyle 4.3596... \cdot 5 = 21.7963\)

This is the final answer, so the entire decimal is no longer needed. Rounded, the perimeter is \(\displaystyle 21.8 \:cm\).

Write the formula to find the perimeter of a pentagon.

\(\displaystyle P=5s\)

Substitute the side length \(\displaystyle 2a+b\) into the equation and simplify.

\(\displaystyle P=5(2a+b)=10a+5b\)