Nonagon

What Is a Nonagon?

A nonagon is a two-dimensional polygon with nine straight sides and nine vertices (corners). The name "nonagon" is a hybrid of a Latin prefix, nonus (meaning "ninth"), and a Greek suffix, gonia (meaning "angle"). An alternative, less common name is "enneagon," which is purely Greek-derived (from ennea, meaning "nine"). In geometry, you'll find that "nonagon" is the more standard term.

Like all polygons, nonagons can be classified into several categories:

• Regular Nonagon: This is the type of nonagon most people picture. In a regular nonagon, all nine sides have the same length, and all nine interior angles are equal in measure. It possesses a high degree of symmetry.
• Irregular Nonagon: An irregular nonagon is any nine-sided polygon that is not regular. This means its sides can have different lengths, and its interior angles can have different measures. The shape can be stretched, skewed, or distorted in countless ways.
• Convex Nonagon: A nonagon is convex if all its interior angles are less than ${180}^{\circ }$. If you were to draw a line connecting any two vertices, the line would lie entirely inside the nonagon. All regular nonagons are convex.
• Concave Nonagon: A nonagon is concave if at least one of its interior angles is greater than ${180}^{\circ }$ (a reflex angle). This creates a "dent" or indentation in the shape. A line connecting two vertices might pass outside the polygon.

Throughout this lesson, we will focus primarily on the properties and calculations related to the regular convex nonagon, as its predictable structure allows for consistent formulas and analysis. However, it's important to remember that the term "nonagon" encompasses a vast family of nine-sided shapes.

What Are the Key Properties of a Regular Nonagon?

A regular nonagon is a beautifully symmetric shape defined by a specific set of properties. Understanding these characteristics is the foundation for performing calculations related to its angles, area, and other geometric features.

Here are the defining properties of a regular nonagon:

• Equal Sides: It has $9$ sides of equal length. If one side measures $5$ cm, all nine sides measure $5$ cm.
• Equal Interior Angles: It has $9$ interior angles of equal measure. As we will calculate later, each interior angle is precisely ${140}^{\circ }$.
• Equal Exterior Angles: It has $9$ exterior angles of equal measure, each measuring ${40}^{\circ }$.
• Rotational Symmetry: A regular nonagon has rotational symmetry of order $9$. This means it can be rotated around its center by multiples of ${40}^{\circ }$ (${360}^{\circ }/9$) and appear unchanged.
• Reflectional Symmetry: It has $9$ lines of symmetry. Each line passes through a vertex and the midpoint of the opposite side.
• Cyclic Property: All vertices of a regular nonagon lie on a single circle, known as the circumscribed circle or circumcircle.

In advanced geometry, polygons are sometimes described using a Schläfli symbol, which is written as $\{p\}$ for a regular polygon with $p$ sides. Therefore, the Schläfli symbol for a regular nonagon is $\{9\}$.

How Do You Calculate the Angles of a Nonagon?

Calculating the angles of a nonagon is a straightforward process once you know the general formulas for any polygon. These formulas rely on the number of sides, which we denote with the variable $n$. For a nonagon, $n=9$.

Sum of the Interior Angles

The first step is often to find the sum of all interior angles. This can be done by dividing the polygon into triangles. By drawing diagonals from one vertex to all other non-adjacent vertices, you can split any $n$-sided polygon into $(n-2)$ triangles. Since the sum of angles in any triangle is ${180}^{\circ }$, we get the following formula:

For a nonagon, where $n=9$, the calculation is:

This means that for any convex nonagon, whether regular or irregular, the measures of its nine interior angles will always add up to ${1260}^{\circ }$.

Single Interior Angle of a Regular Nonagon

If we are dealing with a regular nonagon, all nine interior angles are equal. To find the measure of a single angle, we simply take the total sum and divide by the number of angles ($n$).

For our regular nonagon:

So, every corner of a perfect, regular nonagon measures ${140}^{\circ }$.

Exterior Angles

An exterior angle is formed by extending one side of the polygon and measuring the angle between that extension and the adjacent side. A remarkable property of all convex polygons is that the sum of their exterior angles is always ${360}^{\circ }$.

For a regular nonagon, where all exterior angles are equal, we can find the measure of one by dividing ${360}^{\circ }$ by the number of sides.

The calculation is:

Notice that an interior angle and its corresponding exterior angle form a straight line, so they are supplementary. We can check our work: ${140}^{\circ }$ (interior) + ${40}^{\circ }$ (exterior) = ${180}^{\circ }$.

How Many Diagonals Does a Nonagon Have?

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. Counting them one by one in a shape like a nonagon would be tedious and prone to error. Fortunately, there is a simple and reliable formula to determine the number of diagonals in any polygon.

Let's think about how the formula is derived. A nonagon has $9$ vertices. From any single vertex, you can draw a line to every other vertex.

• You cannot draw a diagonal to the vertex itself.
• You cannot draw a diagonal to its two immediate neighbors (as these lines form the sides of the polygon).

This means that from each vertex, you can draw a diagonal to $n-3$ other vertices. For a nonagon, that's $9-3=6$ diagonals from each vertex.

If we simply multiply the number of vertices ($n$) by the number of diagonals per vertex ($n-3$), we get $n(n-3)$. However, this method counts every diagonal twice (once from each endpoint). For example, the diagonal from vertex A to vertex D is the same as the diagonal from vertex D to vertex A. To correct for this double-counting, we must divide the result by $2$.

This gives us the general formula for the number of diagonals ($D$) in a polygon with $n$ sides:

Now, let's apply this formula to the nonagon, where $n=9$:

A nonagon has exactly $27$ diagonals. This network of internal lines creates a complex and interesting pattern within the shape, dividing it into numerous smaller triangles.

How Do You Find the Area of a Regular Nonagon?

Calculating the area of an irregular nonagon requires complex methods, often by dividing it into triangles. However, for a regular nonagon, we can use specific formulas. The most common methods involve knowing either the side length ($s$) or the apothem ($a$).

Area Using Side Length and Apothem

The apothem is the distance from the center of a regular polygon to the midpoint of one of its sides. It is perpendicular to the side. A regular nonagon can be divided into $9$ congruent isosceles triangles, with the apothem serving as the height of each triangle.

The area of one of these triangles is $\frac{1}{2}\times \text{base}\times \text{height}$, which is $\frac{1}{2}\times s\times a$.

Since there are $9$ such triangles, the total area of the nonagon is:

This can also be expressed using the perimeter ($P=9s$):

Area Using Only the Side Length

In many problems, you are only given the side length ($s$). To find the area, you first need to calculate the apothem using trigonometry. Let's look at one of the $9$ central triangles. If we bisect it with the apothem, we create a right-angled triangle. The angle at the center of the nonagon is $\frac{{360}^{\circ }}{9}={40}^{\circ }$. The apothem bisects this angle, creating an angle of ${20}^{\circ }$. The side of length $s$ is also bisected, creating a base of $\frac{s}{2}$.

Using the tangent function ($\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}$):

Solving for the apothem ($a$):

Now we can substitute this expression for $a$ back into our area formula $\text{Area}=\frac{9sa}{2}$:

Since $\cot (\theta )=\frac{1}{\tan (\theta )}$, we can write the formula more elegantly:

Using a calculator, $\cot ({20}^{\circ })\approx 2.74748$. So, a practical approximation for the area is:

Can You Construct a Perfect Nonagon?

In classical geometry, "construction" refers to drawing a shape using only an unmarked straightedge and a compass. While we can easily construct shapes like equilateral triangles, squares, and hexagons, constructing a regular nonagon is a different story. The surprising answer is that a perfect regular nonagon cannot be constructed using only a compass and straightedge.

This fact is rooted in advanced mathematics, specifically the Gauss-Wantzel theorem. This theorem states that a regular $n$-sided polygon can be constructed if and only if the odd prime factors of $n$ are distinct Fermat primes. A Fermat prime is a prime number of the form $2^{(2^k)}+1$. The first few are $3,5,17,257,...$.

Let's analyze the number of sides of a nonagon, $n=9$. The prime factorization of $9$ is $3^2$. The prime factor is $3$, which is a Fermat prime. However, the theorem requires the prime factors to be distinct. Because the factor $3$ appears twice, a regular nonagon fails the constructibility test.

Constructing a regular nonagon is equivalent to constructing a ${360}^{\circ }/9={40}^{\circ }$ angle. This, in turn, is related to the famous unsolvable problem of "trisecting an angle." Constructing a nonagon requires trisecting a ${120}^{\circ }$ angle (which is constructible), and angle trisection is not possible with only a compass and straightedge.

Approximation and Modern Methods

While a perfect classical construction is impossible, we can create extremely accurate approximations. Artists and engineers have developed methods that get visually indistinguishable from a true nonagon. Of course, in the modern world, we can easily construct a perfect nonagon using tools like a protractor to measure the ${140}^{\circ }$ interior angles or computer-aided design (CAD) software, which can perform the calculations with perfect precision.

What Are Common Mistakes When Working with Nonagons?

When studying nonagons, students can sometimes fall into a few common traps. Being aware of these pitfalls can help you avoid them and improve your accuracy.

• Applying Regular Formulas to Irregular Shapes: The formulas for a single interior angle (${140}^{\circ }$) and the area ($\frac{9}{4}{s}^2\cot ({20}^{\circ })$) are only valid for regular nonagons. Remember that for an irregular nonagon, you must calculate each angle or area component individually.
• Confusing Interior and Exterior Angles: It's easy to mix up the formulas. The sum of interior angles is $(n-2)\times {180}^{\circ }$, while the sum of exterior angles is always ${360}^{\circ }$. A good check is to remember that an interior angle of a regular nonagon (${140}^{\circ }$) is obtuse, while the exterior angle (${40}^{\circ }$) is acute.
• Forgetting to Divide by 2 for Diagonals: A very common error with the diagonal formula, $\frac{n(n-3)}{2}$, is forgetting the final division by $2$. This leads to an answer that is exactly double the correct one. Remember this step corrects for counting each diagonal from both of its endpoints.
• Mixing Up Radius and Apothem: The circumradius ($R$) goes from the center to a vertex, while the apothem ($a$) goes from the center to the midpoint of a side. These are different lengths and are used in different formulas. Using one in place of the other will lead to incorrect area calculations.
• Calculator Mode Errors: When using trigonometric functions to find the area, ensure your calculator is in the correct mode (degrees or radians). The formulas we've discussed, like $\cot ({20}^{\circ })$, use degrees. Using radians will produce a completely different and incorrect result.

Nonagon at a Glance: A Quick Reference

For quick review or reference, here are the key properties and formulas for a nonagon, with a focus on the regular nonagon.