Pentagonal Pyramid — Definition, Formula & Examples

A pentagonal pyramid is a three-dimensional solid that has a pentagon (five-sided polygon) as its base and five triangular faces that rise from the base edges and meet at a single point called the apex.

A pentagonal pyramid is a polyhedron with six faces: one pentagonal base and five lateral triangular faces sharing a common vertex (the apex). It has 10 edges and 6 vertices, satisfying Euler's formula $V - E + F = 2$ .

Key Formula

V = \frac{1}{3} \cdot A_b \cdot h

Where:

$V$ = Volume of the pentagonal pyramid
$A_b$ = Area of the pentagonal base
$h$ = Perpendicular height from the base to the apex

Worked Example

Problem: Find the volume of a pentagonal pyramid whose base has an area of 60 cm² and whose height is 9 cm.

Step 1: Write the pyramid volume formula.

V = \frac{1}{3} \cdot A_b \cdot h

Step 2: Substitute the given values.

V = \frac{1}{3} \cdot 60 \cdot 9

Step 3: Compute the result.

V = \frac{540}{3} = 180 \text{ cm}^3

Answer: The volume of the pentagonal pyramid is 180 cm³.

Why It Matters

Pentagonal pyramids appear as components of more complex solids—for instance, they cap the faces of an icosahedron in certain constructions. Recognizing them helps you count faces, edges, and vertices on composite shapes in geometry courses.

Common Mistakes

Mistake: Counting faces as 5 instead of 6, forgetting to include the pentagonal base.

Correction: A pentagonal pyramid has 5 triangular lateral faces plus 1 pentagonal base, giving 6 faces total.