A pentagonal pyramid is a pyramid where the base is a pentagon. The figure therefore has a total of six faces.

One pentagonal base and five triangular side faces.

If the base is a regular pentagon and the apex is placed directly above the center, it is called a regular pentagonal pyramid. If the apex is not centered, the pyramid is irregular.

Volume

The volume of a pentagonal pyramid is found by multiplying the area of the base by the height and then dividing by 3:

$$ \large V = \frac{1}{3} \cdot A_{base} \cdot h $$

For a regular pentagon with side length \( \large a \), the area of the base can be calculated as:

$$ \large A_{base} = \frac{5}{4} \cdot a^2 \cdot \cot\left(\frac{\pi}{5}\right) $$

Thus, the volume of a regular pentagonal pyramid is:

$$ \large V = \frac{1}{3} \cdot \frac{5}{4} \cdot a^2 \cdot \cot\left(\frac{\pi}{5}\right) \cdot h $$

Surface area

The surface area consists of the area of the base plus the area of the five triangular side faces:

$$ \large S = A_{base} + A_{sides} $$

If the pyramid is regular, all side faces are congruent, and one can write:

$$ \large S = A_{base} + 5 \cdot A_{triangle} $$

Irregular pentagonal pyramid

In an irregular pentagonal pyramid, the apex is not placed above the center of a regular pentagon. The base can be any pentagon, and the side faces are not necessarily equal.

 The volume is still found as:

$$ \large V = \frac{1}{3} \cdot A_{base} \cdot h $$

The surface area in this case requires calculating the area of each triangular side face separately and adding them together with the area of the base:

$$ \large S = A_{base} + \sum_{i=1}^{5} A_{triangle,i} $$