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:

, where the sum of all the angles around the center of the pentagon sum up to  and the  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  because the pictured right triangle is half of a larger isosceles triangle. 

Therefore, the marked angle is . 

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 .

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  must be multiplied by . 

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.

This is the final answer, so the entire decimal is no longer needed. Rounded, the perimeter is .

Write the formula to find the perimeter of a pentagon.

Substitute the side length  into the equation and simplify.

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

Write the formula to find the perimeter for a pentagon.

Substitute the side into the perimeter formula.

Notice that the pentagon is made up of one rectangle and two right triangles. Start by finding the lengths of  and .

Let  be the length of . Then, it follows that  is the length of .

From the figure, we know that . Since we are working with a rectangle, .

Now, solve for .

So then, .

Now that we know , we can use the Pythagorean theorem to solve for the length of .

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

Next, find the length of  by using the Pythagorean theorem.

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

Make sure to round to  places after the decimal.

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

Start by finding the length of . 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 .

Now, we know that  is three times the length of .

Since we have rectangle , .

Thus, 

Now, we can use cosine to find the length of .

Next, use the Pythagorean Theorem to find the length of .

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

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

Start by finding the length of . 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 .

Now, we know that  is three times the length of .

Since we have rectangle , .

Thus, 

Now, we can use sine to find the length of .

Next, use the Pythagorean Theorem to find the length of .

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