Pentagonal Prism — Definition, Formula & Examples

A pentagonal prism is a three-dimensional shape that has two parallel, congruent pentagons as its bases, connected by five rectangles as lateral faces. It has 7 faces, 15 edges, and 10 vertices.

A pentagonal prism is a polyhedron comprising two congruent, parallel pentagonal bases and five rectangular lateral faces perpendicular to the bases (in a right prism). Its volume equals the area of one pentagonal base multiplied by the height (the perpendicular distance between the bases).

Key Formula

V = B \times h \qquad \text{and} \qquad SA = 2B + Ph

Where:

$V$ = Volume of the prism
$B$ = Area of one pentagonal base
$h$ = Height (perpendicular distance between the two bases)
$SA$ = Total surface area
$P$ = Perimeter of the pentagonal base

Worked Example

Problem: A right pentagonal prism has a regular pentagonal base with side length 6 cm and an apothem of 4.13 cm. The height of the prism is 10 cm. Find the volume and total surface area.

Find the base area: For a regular pentagon, the area is (1/2) × perimeter × apothem. The perimeter is 5 × 6 = 30 cm.

B = \tfrac{1}{2} \times 30 \times 4.13 = 61.95 \text{ cm}^2

Find the volume: Multiply the base area by the height of the prism.

V = 61.95 \times 10 = 619.5 \text{ cm}^3

Find the surface area: Add the areas of both bases and all five rectangular lateral faces.

SA = 2(61.95) + 30 \times 10 = 123.9 + 300 = 423.9 \text{ cm}^2

Answer: The volume is 619.5 cm³ and the total surface area is 423.9 cm².

Why It Matters

Pentagonal prisms appear in architecture, packaging design, and cross-section problems in geometry courses. Understanding this shape strengthens your ability to compute volume and surface area for any prism, since the same formulas apply regardless of the base shape.

Common Mistakes

Mistake: Forgetting to include both bases when calculating surface area.

Correction: A prism always has two congruent bases. The total surface area formula includes 2B (twice the base area), not just B.