# Hexagonal Prism: Faces, Volume, and Surface Area

A hexagonal prism is a 3-D solid with two parallel hexagonal bases joined by six rectangles, giving **8 faces**, **18 edges**, and **12 vertices**. This article defines the hexagonal prism, derives its volume and surface area formulas, shows its net, and works through examples.

## The Shape A Pencil Keeps So it Will Not Roll Off the Desk

A **hexagonal prism** is a three-dimensional solid with **two identical, parallel hexagonal bases** connected by **six rectangular side faces**. It belongs to the wider [prisms](/content/math/geometry/prism/index.html) family.

## Faces, Edges, And Vertices

Counting the parts of a hexagonal prism:

- **Faces: 8.** Two hexagonal bases plus six rectangular sides.
- **Vertices: 12.** Each hexagon has 6 corners, and there are two hexagons.
- **Edges: 18.** Each hexagon contributes 6 edges, and 6 more vertical edges join the top corners to the bottom corners.

These satisfy Euler's formula for solids: \( V - E + F = 2 \) gives \( 12 - 18 + 8 = 2 \).

## The Volume Formula And Where It Comes From

Every prism follows the same rule: volume equals base area times height, where the base area of a regular hexagon of side \( a \) is \( \frac{3\sqrt{3}}{2} a^2 \). Thus, the volume \( V = base\ area \times height \) gives:

\[ V = \frac{3\sqrt{3}}{2} a^2 h \]

## The Surface Area Formula

Surface area is the total of every face. It contributes:
- **Two bases:** \( 2 \times \frac{3\sqrt{3}}{2} a^2 = 3\sqrt{3} a^2 \)
- **Lateral surface area:** \( 6ah \)

Total surface area:

\[ SA = 3\sqrt{3} a^2 + 6ah \]

## Examples Of Hexagonal Prism

### Example 1
**Count the faces, edges, and vertices of a hexagonal prism.**
- Faces: \( 2 + 6 = 8 \)
- Vertices: \( 6 \times 2 = 12 \)
- Edges: \( 12 + 6 = 18 \)

### Example 2
**A regular hexagonal prism has base edge \( a=4 \) cm and height \( h=10 \) cm. Find its volume:**

\[ V = \frac{3\sqrt{3}}{2}(4)^2(10) \approx 415.7 \text{ cm}^3 \]

### Example 3
**Spot the error**: Height and base edge cannot be treated as equal. Correctly using height gives:

\[ V \approx 415.7 \text{ cm}^3 \]

### Example 4
**Find the total surface area with \( a=4 \) cm and \( h=10 \) cm:**

\[ SA = 83.1 + 240 = 323.1 \text{ cm}^2 \]

### Example 5
**Find only the lateral surface area with base edge 5 cm and height 12 cm:**

\[ Lateral \ SA = 6(5)(12) = 360 \text{ cm}^2 \]

### Example 6
**Estimate volume of a hexagonal pencil with base edge 3 mm and height 170 mm:**

\[ V \approx 3975 \text{ mm}^3 \]

## Key Takeaways
- A **hexagonal prism** has two parallel hexagonal bases joined by six rectangles.
- It has 8 faces, 18 edges, and 12 vertices, satisfying \( V - E + F = 2 \).
- Volume is \( V = \frac{3\sqrt{3}}{2} a^2 h \).
- Surface area is \( SA = 3\sqrt{3} a^2 + 6ah \).

## A practical next step
Practice these problems to solidify your understanding.