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.

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

Write the perimeter formula for pentagons and substitute the side length.

$\displaystyle P=5s$

$\displaystyle P=5(5\sqrt5)=25\sqrt5$

Write the formula to find the perimeter for a pentagon.

$\displaystyle P=5s$

Substitute the side into the perimeter formula.

$\displaystyle P=5(\frac{\sqrt2}{10})=\frac{\sqrt2}{2}$

Notice that the pentagon is made up of one rectangle and two right triangles. Start by finding the lengths of $\displaystyle AF$ and $\displaystyle FD$.

Let $\displaystyle x$ be the length of $\displaystyle AF$. Then, it follows that $\displaystyle 3x$ is the length of $\displaystyle FD$.

From the figure, we know that $\displaystyle AF+FD=AD$. Since we are working with a rectangle, $\displaystyle AD=BC=24$.

Now, solve for $\displaystyle x$.

$\displaystyle x+3x=24$

$\displaystyle 4x=24$

$\displaystyle x=6$

So then, $\displaystyle AF=6 \text{ and }FD=18$.

Now that we know $\displaystyle AF=6$, we can use the Pythagorean theorem to solve for the length of $\displaystyle AE$.

$\displaystyle AE=\sqrt{(AF)^2+(FE)^2}$

Plug in the lengths of the legs to find the length of the hypotenuse.

$\displaystyle AE=\sqrt{6^2+4^2}=\sqrt{52}$

Next, find the length of $\displaystyle DE$ by using the Pythagorean theorem.

$\displaystyle DE=\sqrt{(FD)^2+(EF)^2}$

$\displaystyle DE=\sqrt{18^2+4^2}=\sqrt{340}$

Finally, add up all the side lengths of the pentagon to find its perimeter.

$\displaystyle \text{Perimeter}=AB+BC+CD+DE+EA$

$\displaystyle \text{Perimeter}=8+24+8+\sqrt{340}+\sqrt{52}=65.65$

Make sure to round to $\displaystyle 2$ places after the decimal.

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of $\displaystyle AF$. Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of $\displaystyle AF$.

$\displaystyle \tan 38=\frac{9}{AF}$

$\displaystyle AF=\frac{9}{\tan38}=11.52$

Now, we know that $\displaystyle FD$ is three times the length of $\displaystyle AF$.

$\displaystyle FD=3(AF)=3(11.52)=34.56$

Since we have rectangle $\displaystyle ABCD$, $\displaystyle AD=BC$.

Thus, $\displaystyle AF+FD=AD=BC$

$\displaystyle BC=11.52+34.56=46.08$

Now, we can use cosine to find the length of $\displaystyle AE$.

$\displaystyle \sin38=\frac{9}{AE}$

$\displaystyle AE=\frac{9}{\sin38}=14.62$

Next, use the Pythagorean Theorem to find the length of $\displaystyle ED$.

$\displaystyle ED=\sqrt{(EF)^2+(FD)^2}$

$\displaystyle ED=\sqrt{9^2+34.56^2}=35.71$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\displaystyle \text{Perimeter}=AB+BC+CD+DE+EA$

$\displaystyle \text{Perimeter}=25+46.08+25+35.71+14.62=146.41$

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of $\displaystyle AF$. Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of $\displaystyle AF$.

$\displaystyle \tan 26=\frac{13}{AF}$

$\displaystyle AF=\frac{13}{\tan 26}=26.65$

Now, we know that $\displaystyle FD$ is three times the length of $\displaystyle AF$.

$\displaystyle FD=3(AF)=3(26.65)=79.95$

Since we have rectangle $\displaystyle ABCD$, $\displaystyle AD=BC$.

Thus, $\displaystyle AF+FD=AD=BC$

$\displaystyle BC=26.65+79.95=106.60$

Now, we can use sine to find the length of $\displaystyle AE$.

$\displaystyle \sin 26=\frac{13}{AE}$

$\displaystyle AE=\frac{13}{\sin 26}=29.66$

Next, use the Pythagorean Theorem to find the length of $\displaystyle ED$.

$\displaystyle ED=\sqrt{(EF)^2+(FD)^2}$

$\displaystyle ED=\sqrt{13^2+79.95^2}=81.00$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\displaystyle \text{Perimeter}=AB+BC+CD+DE+EA$

$\displaystyle \text{Perimeter}=28+106.60+28+81.00+29.66=273.26$