Question 1170768
Let's break down this problem step by step to find the base and lateral areas of the pentagonal pyramid.

**1. Understand the Geometry**

* **Regular Pentagonal Pyramid:**
    * Base is a regular pentagon (all sides and angles equal).
    * Altitude (height) is the perpendicular distance from the apex to the center of the base.
    * Slant height is the altitude of a lateral face (a triangle).

**2. Visualize the Right Triangle**

Imagine a right triangle formed by:

* The altitude (20 cm)
* The slant height (25 cm)
* The apothem (a) of the pentagon (distance from the center of the pentagon to the midpoint of a side).

Using the Pythagorean theorem:

* a² + altitude² = slant height²
* a² + 20² = 25²
* a² + 400 = 625
* a² = 225
* a = √225 = 15 cm

**3. Find the Side Length of the Pentagon (s)**

* Let's consider a right triangle formed by:
    * The apothem (a = 15 cm)
    * Half of a side (s/2)
    * The radius (r) of the pentagon.

* The central angle of a regular pentagon is 360° / 5 = 72°.
* The angle formed by the apothem and the radius is half of the central angle, which is 36°.

* Using trigonometry (tangent):
    * tan(36°) = (s/2) / a
    * s/2 = a * tan(36°)
    * s/2 = 15 * tan(36°)
    * s/2 ≈ 15 * 0.7265
    * s/2 ≈ 10.8975
    * s ≈ 21.795 cm

**4. Find the Base Area (B)**

* Area of a regular pentagon:
    * B = (1/2) * apothem * perimeter
    * Perimeter (P) = 5 * side length
    * P = 5 * 21.795 ≈ 108.975 cm
    * B = (1/2) * 15 cm * 108.975 cm
    * B ≈ 817.3125 cm²

**5. Find the Lateral Area (LA)**

* Lateral Area = (1/2) * perimeter * slant height
* LA = (1/2) * 108.975 cm * 25 cm
* LA ≈ 1362.1875 cm²

**Rounded Values:**

* Base Area (B) ≈ 817.31 cm²
* Lateral Area (LA) ≈ 1362.19 cm²

**Final Answers:**

* **Base Area:** Approximately 817.31 cm²
* **Lateral Area:** Approximately 1362.19 cm²