You are describing a regular pentagon that is 13 cm tall and has sides of 8 cm. To clarify:

1. Height of the Pentagon (13 cm): The height of a regular pentagon is typically measured as the vertical distance from one vertex (at the top) to the midpoint of the opposite side (the base). This is sometimes referred to as the "apotheme" or "altitude" in specific cases.

2. Side Length (8 cm): Each side of the pentagon measures 8 cm.

Now, let's calculate a few properties of the pentagon:

1. Area of the Pentagon:

The formula for the area $A$ of a regular pentagon, given the side length $s$, is:

$A = \frac{5}{4} \times s^2 \times \cot \left( \frac{\pi}{5} \right)$

We can calculate this for a side length of 8 cm.

2. Perimeter of the Pentagon:

The perimeter $P$ of the pentagon is the sum of its sides:

$P = 5 \times s$

For a side length of 8 cm, the perimeter would be:

$P = 5 \times 8 = 40 \, \text{cm}$

3. Apothem:

The apothem (distance from the center to the midpoint of a side) can be calculated using the formula:

$\text{Apothem} = \frac{s}{2 \times \tan \left( \frac{\pi}{5} \right)}$

Would you like me to calculate the area or any other properties of this pentagon using these formulas?

---

Next Steps:

1. Would you like to calculate the area of the pentagon?
2. Should we find the apothem based on the height of 13 cm?
3. Do you want to know how the height relates to the radius of the circumscribed circle?
4. Would you like to visualize the construction of the pentagon?
5. Do you want to explore the interior angles of this pentagon?

Tip: In a regular pentagon, all sides and angles are equal, and the interior angle of each vertex is 108 degrees.