Octagonal Prism — Definition, Formula & Examples

An octagonal prism is a three-dimensional shape that has two parallel, congruent regular octagons as its bases, connected by eight rectangular faces.

An octagonal prism is a polyhedron with 10 faces (2 octagonal bases and 8 lateral rectangular faces), 24 edges, and 16 vertices, formed by translating a regular octagon along an axis perpendicular to its plane.

Key Formula

V = 2(1 + \sqrt{2})\, s^2 \cdot h

Where:

$V$ = Volume of the octagonal prism
$s$ = Side length of the regular octagonal base
$h$ = Height (distance between the two bases)
$2(1+\sqrt{2})\,s^2$ = Area of a regular octagon with side length s

How It Works

To find the volume of an octagonal prism, multiply the area of one octagonal base by the height (the perpendicular distance between the bases). To find the surface area, add the areas of both octagonal bases to the total area of the eight rectangular lateral faces. For a regular octagon with side length

s

, there is a standard formula for its area that you can plug in directly.

Worked Example

Problem: Find the volume of a regular octagonal prism with a base side length of 5 cm and a height of 10 cm.

Step 1: Find the area of the regular octagonal base using the formula.

A = 2(1 + \sqrt{2})\,s^2 = 2(1 + 1.414)(25) = 2(2.414)(25) = 120.7 \text{ cm}^2

Step 2: Multiply the base area by the height to get the volume.

V = A \cdot h = 120.7 \times 10 = 1{,}207 \text{ cm}^3

Answer: The volume is approximately 1,207 cm³.

Why It Matters

Octagonal prisms appear in architecture and engineering — think of octagonal columns or stop-sign-shaped structures extended into 3D. Practicing with them strengthens your ability to apply base-area-times-height reasoning to any prism shape.

Common Mistakes

Mistake: Counting the faces as 8 instead of 10.

Correction: An octagonal prism has 8 rectangular lateral faces plus 2 octagonal bases, giving 10 faces total.