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Hexagonal Prism

Q: Do I have to memorize the formula for the area of a hexagon?

Memorizing the formula $A = \frac{3\sqrt{3}}{2}s^2$ is very helpful and will save you time. However, it's also important to understand where it comes from: it's simply the area of one equilateral triangle ($\frac{\sqrt{3}}{4}s^2$) multiplied by six.

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Ever wondered about the geometry of a honeycomb or a brand new pencil? You're looking at hexagonal prisms! This guide will unlock the secrets of this fascinating 3D shape, showing you exactly how to calculate its surface area and volume with confidence.

Hexagonal Prism — an original Algebra911 reference diagram defining hexagonal prism with its key formula and a worked example. — Hexagonal Prism: Your Complete Guide to Surface Area and Volume

What Is a Hexagonal Prism?

A hexagonal prism is a three-dimensional shape that consists of two parallel hexagonal bases and six rectangular faces connecting the corresponding sides of the bases. Imagine a hexagon, which is a six-sided polygon. Now, picture a second, identical hexagon floating directly above the first one. If you connect each vertex of the bottom hexagon to the corresponding vertex on the top one, you form a hexagonal prism. Because it has a total of eight faces, it is also a type of octahedron, though that term more commonly refers to the Platonic solid with eight triangular faces.

There are two main types of hexagonal prisms:

Right Hexagonal Prism: This is the most common type you'll study. In a right prism, the six side faces are rectangles and are perpendicular to the two hexagonal bases. The shape stands up straight. All the formulas and examples in this guide focus on right regular hexagonal prisms.
Oblique Hexagonal Prism: In this type, the prism appears to be leaning over. The side faces are parallelograms, not rectangles, because they are not perpendicular to the bases.

For our purposes, when we say "hexagonal prism," we will be referring to a right regular hexagonal prism, which means its bases are regular hexagons (all sides and angles are equal).

What Are the Key Properties of a Hexagonal Prism?

Understanding the basic components of a hexagonal prism makes it much easier to work with. Every polyhedron, including our prism, is defined by its faces, edges, and vertices. Let's break them down for a hexagonal prism.

Faces: These are the flat surfaces of the shape. A hexagonal prism has a total of $8$ faces: the $2$ hexagonal bases (the top and bottom) and the $6$ rectangular faces that form the sides.
Edges: These are the straight lines where two faces meet. A hexagonal prism has $18$ edges. You can count them: $6$ edges around the top base, $6$ edges around the bottom base, and $6$ vertical edges connecting the top and bottom.
Vertices: These are the corners where the edges meet. A hexagonal prism has $12$ vertices: $6$ on the top hexagon and $6$ on the bottom hexagon.

Here is a summary of these properties in a table:

Property	Count	Description
Faces	$8$	$2$ hexagonal bases and $6$ rectangular lateral faces.
Edges	$18$	$6$ on the top base, $6$ on the bottom base, and $6$ vertical edges.
Vertices	$12$	$6$ on the top base and $6$ on the bottom base.

These properties are consistent for all hexagonal prisms, whether they are right, oblique, regular, or irregular. However, to perform calculations for area and volume, we almost always work with right regular hexagonal prisms, where the calculations are predictable.

How to Find the Area of the Hexagonal Base

Before we can find the total surface area or volume of the prism, we must first master finding the area of its hexagonal base. A regular hexagon is a special shape because it can be divided into six identical equilateral triangles, all meeting at the center. This is the key to finding its area!

Let's denote the side length of the regular hexagon as $s$ . Since the hexagon is made of six equilateral triangles, each of these triangles also has a side length of $s$ .

Area of one equilateral triangle: The formula for the area of an equilateral triangle with side length $s$ is $\frac{\sqrt{3}}{4} s^{2}$ .
Area of the hexagon: Since the hexagon is composed of six of these triangles, we simply multiply the area of one triangle by $6$ .

Area of Hexagon $= 6 \times (\frac{\sqrt{3}}{4} s^{2}) = \frac{6 \sqrt{3}}{4} s^{2}$

Simplifying the fraction $\frac{6}{4}$ to $\frac{3}{2}$ , we get the final formula for the area of the base, which we'll call $A_{b a s e}$ .

A_{base} = \frac{3\sqrt{3}}{2}s^2

Remember, $s$ is the length of one side of the regular hexagonal base. You will often use a calculator for $\sqrt{3}$ , which is approximately $1.732$ .

Example 1

Find the area of a regular hexagon with a side length of $4$ cm.

Solution:
We are given the side length $s = 4$ cm.
We use the formula for the area of a regular hexagon:
$A_{b a s e} = \frac{3 \sqrt{3}}{2} s^{2}$
Substitute $s = 4$ into the formula:
$A_{b a s e} = \frac{3 \sqrt{3}}{2} (4)^{2}$
$A_{b a s e} = \frac{3 \sqrt{3}}{2} (16)$
Now, we can simplify the calculation:
$A_{b a s e} = 3 \sqrt{3} \times 8$
$A_{b a s e} = 24 \sqrt{3} {cm}^{2}$
For a decimal approximation, we use $\sqrt{3} \approx 1.732$ :
$A_{b a s e} \approx 24 \times 1.732 \approx 41.57 {cm}^{2}$
The area of the hexagonal base is $24 \sqrt{3}$ cm², or approximately $41.57$ cm².

How Do You Calculate the Surface Area of a Hexagonal Prism?

The surface area of a 3D object is the total area of all its faces combined. For a hexagonal prism, this means we need to find the area of the two hexagonal bases and add it to the area of the six rectangular sides. We can break this down into two parts:

The Area of the Bases: We have two identical hexagonal bases (a top and a bottom). We already know the formula for the area of one base is $A_{b a s e} = \frac{3 \sqrt{3}}{2} s^{2}$ . So, the area for both bases is just $2 \times A_{b a s e}$ .
The Lateral Area: This is the combined area of the six rectangular faces on the sides. Imagine unrolling these six rectangles and laying them flat. They would form one large rectangle! The height of this large rectangle is the height of the prism, $h$ . The width is the perimeter of the hexagonal base. Since the base has $6$ sides of length $s$ , its perimeter is $6 s$ . Therefore, the lateral area ( $A_{l a t e r a l}$ ) is $(6 s) \times h$ .

To get the Total Surface Area (SA), we add these two parts together:

$S A = (Area of 2 Bases) + (Lateral Area)$

$S A = 2 (\frac{3 \sqrt{3}}{2} s^{2}) + (6 s h)$

Simplifying the first term gives us the final, powerful formula:

SA = 3\sqrt{3}s^2 + 6sh

Here, $s$ is the side length of the hexagonal base and $h$ is the height of the prism.

Example 2

Calculate the total surface area of a hexagonal prism with a base side length of $5$ inches and a height of $10$ inches.

Solution:
We are given $s = 5$ inches and $h = 10$ inches.
We will use the surface area formula: $S A = 3 \sqrt{3} s^{2} + 6 s h$ .
First, let's calculate the area of the two bases: $3 \sqrt{3} s^{2}$ .
$Area of Bases = 3 \sqrt{3} (5)^{2} = 3 \sqrt{3} (25) = 75 \sqrt{3} {in}^{2}$
Next, let's calculate the lateral area: $6 s h$ .
$Lateral Area = 6 (5) (10) = 300 {in}^{2}$
Finally, add them together for the total surface area:
$S A = 75 \sqrt{3} + 300 {in}^{2}$
To get a numerical answer, use $\sqrt{3} \approx 1.732$ :
$S A \approx 75 (1.732) + 300$
$S A \approx 129.9 + 300 = 429.9 {in}^{2}$
The total surface area is $300 + 75 \sqrt{3}$ in², or approximately $429.9$ in².

How Do You Calculate the Volume of a Hexagonal Prism?

Calculating the volume of a prism is often simpler than finding its surface area. The volume of any right prism, no matter the shape of its base, follows a single, elegant rule: the area of the base multiplied by the height of the prism.

Think of it like stacking identical, super-thin hexagonal sheets on top of each other until you reach the height $h$ . The volume is the total space occupied by all those sheets.

The general formula for a prism's volume is:

$V = A_{b a s e} \times h$

We already have our specific formula for the area of a regular hexagonal base: $A_{b a s e} = \frac{3 \sqrt{3}}{2} s^{2}$ . All we have to do is substitute this into the general volume formula.

V = \left( \frac{3\sqrt{3}}{2}s^2 \right) h

This formula allows you to find the volume if you know the side length of the hexagonal base ( $s$ ) and the height of the prism ( $h$ ). Remember, volume is measured in cubic units, like cm³ or in³.

Example 3

A hexagonal prism has a base side length of $2$ meters and a height of $7$ meters. What is its volume?

Solution:
We are given $s = 2$ m and $h = 7$ m.
We use the volume formula: $V = (\frac{3 \sqrt{3}}{2} s^{2}) h$ .
First, let's find the area of the base, $A_{b a s e}$ :
$A_{b a s e} = \frac{3 \sqrt{3}}{2} (2)^{2} = \frac{3 \sqrt{3}}{2} (4) = 6 \sqrt{3} m^{2}$
Now, multiply the base area by the height to find the volume:
$V = A_{b a s e} \times h = (6 \sqrt{3}) \times 7$
$V = 42 \sqrt{3} m^{3}$
For a decimal approximation, use $\sqrt{3} \approx 1.732$ :
$V \approx 42 \times 1.732 \approx 72.744 m^{3}$
The volume of the prism is $42 \sqrt{3}$ m³, or approximately $72.74$ m³.

Key formulas for hexagonal prism by Algebra911.

What Are Some Common Mistakes to Avoid?

Working with hexagonal prisms can be tricky, and a few common errors can trip students up. Being aware of these pitfalls is the first step to avoiding them!

Confusing Surface Area and Volume: This is a fundamental mistake. Remember, area is a measure of flat, 2D space (measured in units²) like the paper needed to wrap the prism. Volume is a measure of the 3D space inside (measured in units³) like the amount of water the prism can hold. Their formulas and units are completely different.
Forgetting to Double the Base Area: When calculating total surface area, a very common error is to calculate the lateral area and add the area of only one base. A prism always has two bases (a top and a bottom), so you must multiply your $A_{b a s e}$ by $2$ . Our final formula $S A = 3 \sqrt{3} s^{2} + 6 s h$ already does this for you, as $3 \sqrt{3} s^{2} = 2 \times \frac{3 \sqrt{3}}{2} s^{2}$ .
Mixing up Side Length ( $s$ ) and Height ( $h$ ): Always double-check which number is which. The side length $s$ is a measurement on the flat hexagonal base. The height $h$ is the distance between the two bases. Plugging them into the wrong spots in the formula will lead to an incorrect answer.
Calculation Errors with $\sqrt{3}$ : The $\sqrt{3}$ part of the formula can be intimidating. Don't panic! It's just a number (approximately $1.732$ ). It's often best to leave $\sqrt{3}$ in your answer for an exact form (like $24 \sqrt{3}$ ) until the very last step. Then, if required, use a calculator to get a decimal approximation. Be sure to follow any rounding instructions given in the problem.
Using Radius or Apothem Instead of Side Length: Some problems might give you the apothem (the distance from the center to the midpoint of a side) or the radius (the distance from the center to a vertex). The formulas we've used here are specifically for when you are given the side length $s$ . If you are given a different measurement, you would need to first use it to find $s$ before proceeding.

Quick Summary and Formula Reference

This lesson covered a lot of ground. Here is a quick reference guide to the key concepts and formulas for a right regular hexagonal prism. You can use this as a study aid to quickly review the most important information.

Key Properties:

Faces: $8$ ( $2$ hexagons, $6$ rectangles)
Edges: $18$
Vertices: $12$

Key Variables:

$s$ : The length of one side of the regular hexagonal base.
$h$ : The height of the prism (the distance between the two bases).

Formula Reference Table:

Calculation	Formula
Area of one Base ( $A_{b a s e}$ )	$\frac{3 \sqrt{3}}{2} s^{2}$
Lateral Surface Area ( $A_{l a t e r a l}$ )	$6 s h$
Total Surface Area ( $S A$ )	$3 \sqrt{3} s^{2} + 6 s h$
Volume ( $V$ )	$(\frac{3 \sqrt{3}}{2} s^{2}) h$

The best way to get comfortable with these formulas is to practice. Work through the examples again and try some problems on your own. By breaking down each problem into smaller steps—finding the base area first, then the lateral area, then combining them—you can solve even the most complex questions about hexagonal prisms.

Frequently Asked Questions

What is the difference between a hexagonal prism and a hexagonal pyramid?

A hexagonal prism has two hexagonal bases and rectangular side faces, making it look like a box with a hexagonal profile. A hexagonal pyramid has only one hexagonal base and its side faces are triangles that meet at a single point (the apex).

Where can you find hexagonal prisms in the real world?

Hexagonal prisms are surprisingly common! A classic example is a new, unsharpened pencil. You can also see them in the shape of metal nuts, some bolts, and most famously, in the structure of a honeycomb made by bees.

What does the 's' in the formulas stand for?

The variable 's' stands for the side length of the regular hexagonal base. It's the length of one of the six equal sides that make up the hexagon at the top and bottom of the prism.

What does the 'h' in the formulas stand for?

The variable 'h' represents the height of the prism. This is the perpendicular distance between the two hexagonal bases. It tells you how 'tall' the prism is.

Can the side faces of a hexagonal prism be squares?

Yes, they can. The side faces are rectangles, and a square is a special type of rectangle. This would happen if the height of the prism (h) is exactly equal to the side length of the hexagonal base (s).

Do I have to memorize the formula for the area of a hexagon?

Memorizing the formula $A = \frac{3 \sqrt{3}}{2} s^{2}$ is very helpful and will save you time. However, it's also important to understand where it comes from: it's simply the area of one equilateral triangle ( $\frac{\sqrt{3}}{4} s^{2}$ ) multiplied by six.

What is a 'regular' hexagonal prism?

A regular hexagonal prism is a prism whose bases are regular hexagons. A regular hexagon is a six-sided shape where all six sides have the same length and all six interior angles are equal. The formulas in this guide apply specifically to right regular hexagonal prisms.