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Nonagon – Definition with Examples

Jo-ann Caballes

Updated on January 11, 2024

Welcome to Brighterly! Today, we are venturing into the captivating realm of geometry, with a particular emphasis on the intriguing nine-sided polygon known as the Nonagon. At Brighterly, our goal is to ignite your curiosity and inspire a love for learning. Together, let’s explore the remarkable features, angles, and real-world applications of nonagons, and discover how they can spark our creativity and enhance our understanding of the world around us!

What is a Nonagon?

A Nonagon is a captivating polygon that consists of exactly nine sides and nine vertices. These intriguing nine-sided figures can manifest in various forms, such as regular and irregular shapes, as well as both convex and concave configurations. Nonagons are prominent in a wide range of areas, spanning from the realms of art and design to the fields of mathematics and engineering. Their unique structure and fascinating properties make them an essential shape to explore when learning about geometry.

Nonagon Sides

The sides of a nonagon can exhibit different lengths, which is dependent on whether it is a regular or an irregular nonagon. In a regular nonagon, all of the sides maintain equal lengths, creating a harmonious and balanced shape. In contrast, an irregular nonagon displays sides with varying lengths, leading to a more asymmetrical and unique form. These diverse characteristics allow nonagons to be used in a multitude of ways and applications, from creative designs to problem-solving in mathematics.

Regular Nonagon

A regular nonagon is a distinctive type of nonagon in which all sides are congruent in length, and all angles are equal in measure. This symmetrical shape exhibits an interior angle sum of 1,260°, with each individual interior angle measuring 140°. The regular nonagon’s consistency in both its angles and side lengths results in a visually appealing and mathematically significant polygon that can be applied to various disciplines.

Properties of a Regular Nonagon

A regular nonagon possesses several unique properties that set it apart from other polygons:

• Equilateral and equiangular: All of the sides and angles are equal, making the nonagon both equilateral and equiangular.
• Symmetry: The nonagon has 9 lines of symmetry, which contributes to its balanced and harmonious appearance.
• Interior angle sum: The sum of the interior angles in a regular nonagon is 1,260°, which is a key characteristic that helps distinguish it from other polygons.

Convex Nonagon and Concave Nonagon

When it comes to nonagons, there are two primary classifications based on their shape: convex and concave. Each type exhibits its own unique characteristics and aesthetic appeal.

A convex nonagon is a nine-sided polygon in which all of its interior angles measure less than 180°, and all its vertices point outwards. The result is a shape that appears to “bulge” outwards, providing a sense of fullness and stability.

On the other hand, a concave nonagon has at least one interior angle measuring more than 180°, which creates an indentation in the shape. This “caved-in” appearance gives concave nonagons a more dynamic and complex visual presence, making them an intriguing choice for various artistic and mathematical applications.

Nonagon Angles

Nonagons feature nine angles formed by the intersection of their sides, which contribute to their unique geometric structure. The sum of these angles relies on whether the nonagon is regular or irregular, as each category exhibits its own specific characteristics and properties.

Nonagon Interior Angles

The interior angles of a nonagon are the angles formed between the sides within the shape itself. In a regular nonagon, all interior angles are equal, measuring a consistent 140°. This uniformity contributes to the regular nonagon’s harmonious and symmetrical appearance, while irregular nonagons display varying interior angle measurements.

Sum of Interior Angles of a Nonagon

To calculate the sum of the interior angles of a nonagon, use the following formula:

Sum of Interior Angles = (n - 2) × 180°

In this equation, n represents the number of sides in the polygon. For a nonagon, n = 9:

Sum of Interior Angles = (9 - 2) × 180° = 7 × 180° = 1,260°

Regardless of whether the nonagon is regular or irregular, the sum of its interior angles always equates to 1,260°.

Nonagon Exterior Angles

The exterior angles of a nonagon are the angles formed outside the shape between a side and an adjacent side extended. In a regular nonagon, all exterior angles are equal, measuring a consistent 40°. These angles play a crucial role in determining the overall appearance and properties of the nonagon.

Properties of Nonagon

Nonagons boast several unique properties that set them apart from other polygons:

• A nonagon has nine sides and nine vertices.
• It can be regular or irregular, convex, or concave.
• The sum of the interior angles of a nonagon always amounts to 1,260°.

Nonagon Diagonals

In geometry, a diagonal is a line segment that connects two non-adjacent vertices within a polygon. In a nonagon, each vertex can connect to six other vertices, resulting in a total of 9 × 6 / 2 = 27 diagonals. These diagonals provide additional structural and visual complexity to the nonagon shape, further enhancing its mathematical and artistic appeal.

Perimeter of Nonagon

Calculating the perimeter of a nonagon involves summing up the lengths of its nine sides. For a regular nonagon, where all side lengths are equal (s), the calculation becomes even simpler by utilizing the following formula:

Perimeter = 9 × s

In this way, you can easily determine the perimeter of any regular nonagon by simply multiplying its side length by 9.

Examples on Nonagon

Let’s explore some examples to better understand the various properties of nonagons:

1. A regular nonagon with a side length of 5 cm:

   • Perimeter: 9 × 5 = 45 cm
   • Sum of interior angles: 1,260°
   • Each interior angle: 140°
   • Each exterior angle: 40°
   • Number of diagonals: 27

2. An irregular convex nonagon with varying side lengths:

   • The perimeter will vary depending on the side lengths.
   • The sum of the interior angles remains constant at 1,260°.
   • The individual interior angles will vary depending on the shape, leading to a more diverse and visually complex figure.

Practice Questions on Nonagon

Enhance your understanding of nonagons by answering the following practice questions:

1. What is the perimeter of a regular nonagon with a side length of 8 cm?
2. How many diagonals does a nonagon have?
3. What is the measure of an exterior angle of a regular nonagon?

By engaging with these questions, you’ll gain a deeper appreciation for the unique properties and characteristics of nonagons, solidifying your knowledge of this fascinating polygon.

Frequently Asked Questions on Nonagon

What is the difference between a regular and an irregular nonagon?

A regular nonagon is a nine-sided polygon with all sides and angles equal. This results in a symmetrical, balanced shape with a harmonious appearance. In contrast, an irregular nonagon has varying side lengths and angle measures, leading to a more diverse and visually complex figure. Irregular nonagons can be either convex or concave, depending on the arrangement of their angles and sides.

How many lines of symmetry does a regular nonagon have?

A regular nonagon has 9 lines of symmetry. These lines of symmetry contribute to the shape’s balanced and harmonious appearance, which is a key characteristic of regular polygons. In general, a regular polygon with n sides will have n lines of symmetry.

What are some real-life examples of nonagons?

Nonagons can be found in a variety of real-life contexts, showcasing their versatility and unique properties. Some examples include:

• Art and design: Nonagons may appear in tiling patterns, mosaics, and decorative elements. Their unique shape and symmetrical properties make them an attractive choice for artists and designers.
• Architecture: Nonagonal structures or architectural features can be found in some buildings, such as churches or monuments. These elements can add visual interest and structural complexity to the design.
• Mathematics: Nonagons serve as a fundamental shape in the study of geometry and polygonal mathematics. They are used to explore concepts such as perimeter, area, angles, and diagonals.
• Games and puzzles: Nonagons can be found in various puzzles and games, like geometric dissection puzzles or pattern-matching games, where players need to arrange or manipulate shapes to achieve a specific goal.

By understanding the various real-life applications of nonagons, you can better appreciate the importance and versatility of this intriguing nine-sided polygon.

Jo-ann Caballes

5 articles

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.