SOLUTION: A regular pentagonal pyramid has an altitude of 20 cm and a slant height which measures 25cm. Find the base and the lateral areas of the pyramid.

Question 1170768: A regular pentagonal pyramid has an altitude of 20 cm and a slant height which measures 25cm. Find the base and the lateral areas of the pyramid.
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
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

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