Octagon

What Exactly Is an Octagon?

An octagon is a two-dimensional geometric shape that is defined as a polygon with exactly eight sides, eight vertices (corners), and eight interior angles. The name itself gives a clue to its identity, deriving from the Greek words 'oktō' meaning 'eight' and 'gōnia' meaning 'angle.' While the most famous octagon is the one you see on a stop sign, octagons come in many forms.

We can classify octagons into several types:

• Regular Octagon: This is the most symmetrical type. In a regular octagon, all eight sides are equal in length, and all eight interior angles are equal in measure. The stop sign is a classic example of a regular octagon.
• Irregular Octagon: An irregular octagon is any eight-sided polygon where the sides are not all equal in length or the angles are not all equal. Their shapes can be quite varied and unpredictable.
• Convex Octagon: In a convex octagon, all interior angles are less than ${180}^{\circ }$. All the vertices point outwards, away from the center of the shape. A regular octagon is always convex.
• Concave Octagon: A concave octagon has at least one interior angle that is greater than ${180}^{\circ }$. This means at least one vertex 'caves in' or points toward the interior of the polygon.

For the majority of our work in geometry, especially at this level, our focus will be on the regular convex octagon because its properties are predictable and consistent, allowing us to develop powerful formulas.

What Are the Key Properties of a Regular Octagon?

Regular octagons are full of interesting and consistent properties that we can calculate. Because they are symmetrical, we can determine their angles with precision. Let's explore the fundamental geometric characteristics of a regular octagon.

Sum of Interior Angles

For any convex polygon, we can find the sum of its interior angles using a single, reliable formula. The formula depends on the number of sides, which we'll call $n$.

For an octagon, $n=8$. Plugging this into the formula:

Sum = $(8-2)\times {180}^{\circ }=6\times {180}^{\circ }={1080}^{\circ }$

So, the sum of all interior angles in any octagon, regular or irregular, is always ${1080}^{\circ }$.

Measure of a Single Interior Angle

Since a regular octagon has eight equal angles, we can find the measure of a single angle by dividing the total sum by the number of angles (which is 8).

Every corner you see on a regular octagon forms a ${135}^{\circ }$ angle.

Measure of a Single Exterior Angle

The exterior angle is the angle formed by extending one side of the polygon and the adjacent side. For any regular polygon, the sum of the exterior angles is always ${360}^{\circ }$. To find a single exterior angle of a regular octagon, we simply divide by 8.

Notice that the interior angle (${135}^{\circ }$) and the exterior angle (${45}^{\circ }$) add up to ${180}^{\circ }$. This is always true for any polygon, as they form a linear pair.

How Do You Calculate the Perimeter of an Octagon?

The perimeter of any polygon is simply the total distance around its exterior, which you find by adding the lengths of all its sides. This concept applies to both regular and irregular octagons, though the calculation is much simpler for regular ones.

Irregular Octagon:

To find the perimeter of an irregular octagon, you must know the length of each of its eight different sides. If the side lengths are $s_1,s_2,s_3,s_4,s_5,s_6,s_7,$ and $s_8$, the perimeter $P$ is:

Regular Octagon:

For a regular octagon, life is much easier! Since all eight sides are equal in length, we can just call the length of one side $s$. The perimeter is found by multiplying the side length by 8.

How Do You Find the Area of a Regular Octagon?

Calculating the area of a regular octagon is more complex than finding its perimeter, but it can be done with a powerful formula. While there are methods involving trigonometry and dividing the shape into triangles (which we'll cover next), a direct formula exists if you know the length of a side, $s$.

The formula might look intimidating at first, but it's just a matter of plugging in the side length. It involves the square root of 2, which is a common number in geometry related to ${45}^{\circ }$ angles—the same as the octagon's exterior angle!

Here, $s$ is the length of one side of the regular octagon. The value of $\sqrt{2}$ is approximately $1.414$. So, the formula can also be approximated as:

This approximation is very useful for getting a quick numerical answer without having to leave $\sqrt{2}$ in your result.

Can You Find the Area by Deconstructing the Octagon?

Yes, and it's a fantastic way to understand where the area formula comes from! A regular octagon can be perfectly divided into 8 congruent (identical) isosceles triangles, with their vertices meeting at the center of the octagon. By finding the area of one of these triangles and multiplying it by 8, we can find the total area of the octagon.

Imagine lines drawn from the center of the octagon to each of its 8 vertices. You now have 8 triangles. Let's analyze one of them:

• The two equal sides of the triangle are equal to the radius ($r$) of the octagon, which is the distance from the center to any vertex.
• The angle between these two sides, at the center of the octagon, is $\frac{{360}^{\circ }}{8}={45}^{\circ }$.

If we know the radius $r$, we can use a variation of the triangle area formula that involves trigonometry: $\text{Area}=\frac{1}{2}ab\sin (C)$, where $C$ is the angle between sides $a$ and $b$. In our case, $a=r$, $b=r$, and $C={45}^{\circ }$.

Area of one triangle = $\frac{1}{2}r\cdot r\cdot \sin ({45}^{\circ })=\frac{1}{2}{r}^2\sin ({45}^{\circ })$.

Since $\sin ({45}^{\circ })=\frac{\sqrt{2}}{2}$, the area of one triangle is $\frac{1}{2}{r}^2\left(\frac{\sqrt{2}}{2}\right)=\frac{\sqrt{2}}{4}{r}^2$.

The total area of the octagon is 8 times this amount:

This gives us a second powerful formula for the area, this time using the radius instead of the side length!

Where Do We See Octagons in the Real World?

The octagon's unique and stable shape makes it useful and visually appealing in many contexts. While some are obvious, others might surprise you.

• Stop Signs: The most universally recognized octagon. Its eight sides make it distinct from other traffic signs, ensuring it is easily identifiable even from a distance or when covered in snow.
• Architecture: Throughout history, architects have used octagons for buildings, towers, and windows. Famous examples include the Dome of the Rock in Jerusalem and the tower of Ely Cathedral in England. The shape provides a wider field of view than a square and a more structured feel than a circle.
• Nuts and Bolts: The heads of many bolts are hexagonal (6-sided), but octagonal nuts are also used in certain applications, providing multiple points for a wrench to grip.
• Flooring and Tiling: Octagonal tiles are often used in decorative flooring patterns, typically paired with small square tiles to fill the gaps, creating a beautiful and classic tessellation.
• Everyday Objects: You can find octagons in the design of some umbrellas, poker tables, gazebos, and even the fighting ring used in the Ultimate Fighting Championship (UFC).

What Are Some Common Mistakes When Working with Octagons?

When dealing with octagons, a few common pitfalls can lead to incorrect answers. Being aware of them is the first step to avoiding them.

• Assuming All Octagons are Regular: The biggest mistake is applying formulas for regular octagons (like $A=2(1+\sqrt{2})s^2$ or interior angle = ${135}^{\circ }$) to an irregular shape. These formulas only work if all sides and angles are equal.
• Confusing Perimeter and Area: It's a fundamental error, but it happens. Remember, perimeter is a measure of length (units like cm, ft, in) while area is a measure of space (units like cm², ft², in²). Always double-check that you're calculating what the question asks for.
• Mixing up Radius and Apothem: The radius goes from the center to a vertex (a corner). The apothem goes from the center to the midpoint of a side. They are not the same length! The radius is always longer than the apothem in any regular polygon.
• Calculation Errors with $\sqrt{2}$: When using the area formula, be careful with the order of operations. Calculate $s^2$ first, then multiply by the term in the parentheses. When approximating, use at least three decimal places ($1.414$) for better accuracy.
• Using Side Length in the Radius Formula: Be careful not to mix the two area formulas. If you have the side length $s$, use $A=2(1+\sqrt{2})s^2$. If you have the radius $r$, use $A=2\sqrt{2}{r}^2$. Using the wrong measurement in a formula will give a wildly incorrect answer.

Quick Summary and Key Formulas

This section provides a quick reference for the key properties and formulas related to the regular octagon. Use it as a study guide or a quick refresher.