**Instructions:**

- Enter the number of sides (n), side length (s), and circumradius (R) of the regular polygon.
- Click "Calculate" to calculate the perimeter, apothem, area, and other parameters.
- Your calculation history will be displayed below the results.
- You can clear the results and history using the "Clear" button.
- Click "Copy" to copy the results to the clipboard.

**Results**

**Calculation History**

## What is Regular Polygon?

A regular polygon is a two-dimensional geometric shape with straight sides and angles where all sides have equal length and all angles have equal measure. In other words, it is a polygon whose sides are congruent (have the same length) and whose interior angles are congruent (have the same measure).

Some common examples of regular polygons include:

- Equilateral Triangle: A three-sided polygon where all sides and angles are congruent.
- Square: A four-sided polygon with all sides of equal length and all angles measuring 90 degrees.
- Pentagon: A five-sided polygon with all sides and angles congruent.
- Hexagon: A six-sided polygon with all sides and angles congruent.
- Octagon: An eight-sided polygon with all sides and angles congruent.

## All Formulae Related to Regular Polygon

- Perimeter (P) of a Regular Polygon: P = n * s Where:
- P is the perimeter of the polygon.
- n is the number of sides of the polygon.
- s is the length of each side.

- Interior Angle (A) of a Regular Polygon: A = (n – 2) * 180° / n Where:
- A is the measure of each interior angle in degrees.
- n is the number of sides of the polygon.

- Exterior Angle (E) of a Regular Polygon: E = 360° / n Where:
- E is the measure of each exterior angle in degrees.
- n is the number of sides of the polygon.

- Apothem (r) of a Regular Polygon: r = (s / 2) * cot(π / n) Where:
- r is the length of the apothem (the distance from the center to the midpoint of a side).
- s is the length of each side.
- n is the number of sides.
- cot(π / n) is the cotangent of half the interior angle.

- Area (A) of a Regular Polygon: A = (1/2) * n * s * r Where:
- A is the area of the polygon.
- n is the number of sides.
- s is the length of each side.
- r is the length of the apothem.

- Circumradius (R) of a Regular Polygon: R = (s / 2) * csc(π / n) Where:
- R is the circumradius (the distance from the center to a vertex).
- s is the length of each side.
- n is the number of sides.
- csc(π / n) is the cosecant of half the interior angle.

## Practical Uses of Regular Polygon

Regular polygons have various practical uses in everyday life, as well as in mathematics, engineering, and other fields. Here are some practical uses of regular polygons:

**Architecture and Construction:**Regular polygons, such as squares and equilateral triangles, are commonly used in architectural design. For example, square tiles are used in flooring, and equilateral triangles can be found in the design of certain roof trusses.**Geometry and Mathematics:**Regular polygons serve as fundamental shapes for studying geometry. They are used in theorems, proofs, and geometric constructions to build more complex shapes and solve problems.**Manufacturing:**Regular polygons are used in manufacturing and machining processes. Engineers and machinists may work with regular shapes when designing and creating objects or components.**Art and Design:**Artists and designers use regular polygons as basic building blocks for creating patterns, tessellations, and artwork. Symmetrical designs based on regular polygons can be aesthetically pleasing.**Computer Graphics:**Regular polygons play a crucial role in computer graphics for rendering 2D and 3D shapes. Algorithms for drawing and manipulating regular polygons are essential in computer game development, computer-aided design (CAD), and computer-generated imagery (CGI).**Navigation and Surveying:**Regular polygons, such as triangles and hexagons, are used in navigation and surveying for measuring angles, distances, and land boundaries.

## Applications of Regular Polygon in Various Fields

Regular polygons find applications in various fields due to their symmetrical and well-defined geometric properties. Here are some specific applications of regular polygons in different fields:

**Geometry and Mathematics:**- Regular polygons serve as fundamental examples for studying geometry and trigonometry.
- They are used in the construction of other geometric shapes and figures.
- Regular polygons are a basis for exploring the relationships between angles, sides, and diagonals.

**Architecture and Construction:**- Regular polygons are used in architectural design to create aesthetically pleasing patterns and layouts.
- They are employed in the design of floor tiles, wall patterns, and decorative elements.
- Regular polygons like squares and rectangles are used in building foundations, walls, and rooms.

**Computer Graphics and Animation:**- Regular polygons are essential in computer graphics for rendering shapes and objects.
- They are used in 3D modeling, where polygons are the building blocks for creating complex 3D structures.
- Regular polygons are used in rendering 2D shapes and sprites in video games.

**Engineering:**- Regular polygons play a role in structural engineering for designing and analyzing geometric aspects of buildings and bridges.
- In civil engineering, regular polygons are used in land surveying for measuring angles and distances.

**Crystallography:**- Regular polygons appear in the atomic structures of crystals.
- Crystallographers use regular polygons to describe the repeating patterns found in crystal lattices.

## Benefits of Using the Regular Polygon

The use of regular polygons offers several benefits in various fields and applications due to their well-defined geometric properties and symmetries. Here are some of the benefits of using regular polygons:

**Simplicity and Predictability:**Regular polygons have equal side lengths and interior angles, making them easy to work with mathematically. This simplicity allows for straightforward calculations and predictions.**Symmetry:**Regular polygons exhibit a high degree of symmetry, making them aesthetically pleasing and useful for creating balanced and harmonious designs in art, architecture, and other creative fields.**Ease of Construction:**Constructing regular polygons is relatively simple using geometric tools and principles, making them ideal for teaching and demonstrating geometric concepts.**Uniformity:**Regular polygons provide a uniform and consistent appearance in designs and patterns, enhancing visual appeal and regularity in various applications.**Tessellation:**Regular polygons can be used to create tessellations, which are repeating patterns with no gaps or overlaps. Tessellations are useful in art, design, and tile work.**Structural Stability:**In architecture and engineering, regular polygons like squares and triangles are used for their structural stability and load-bearing capabilities.**Efficiency:**Regular polygons can be used to efficiently pack objects into a given space, such as hexagons in beehives or squares in a checkerboard pattern.

## References

**“Regular Polygons and Symmetry”**by Marjorie Senechal**“Constructing Regular Polygons with Compass and Straightedge”**by George E. M. Moor

Last Updated : 31 July, 2024

Sandeep Bhandari holds a Bachelor of Engineering in Computers from Thapar University (2006). He has 20 years of experience in the technology field. He has a keen interest in various technical fields, including database systems, computer networks, and programming. You can read more about him on his bio page.

Kyle54The in-depth explanations and practical examples of regular polygons provided in this article have broadened my understanding of their significance and applications.

Murphy PatrickThe link between regular polygons and geometry/mathematics is made very clear in this article, highlighting their foundational role in these subjects.

Kennedy EvaI agree, the geometric properties and relationships discussed here make regular polygons an essential concept in geometry and trigonometry.

Oliver59Regular polygons are an essential part of various fields, and this article effectively highlights their use in architecture, computer graphics, and engineering.

Wayne BellThis is a very informative article about regular polygons. It does a great job of explaining their properties and practical uses.

JleeI’ve always been fascinated by the geometry of regular polygons, and this article provides a comprehensive overview of their applications in various fields.

Colin MorganThe versatility and significance of regular polygons are well-explained in this article, making it a valuable read for anyone interested in geometry or related fields.

Sienna HuntThis article effectively captures the essence of regular polygons and their importance in diverse fields, offering a holistic view of their relevance.

Jane13The practical applications of regular polygons in different fields are impressive, especially their role in computer graphics and engineering.

Butler DavidThe detailed formulas provided for regular polygons are very helpful and add depth to the understanding of their properties.

Russell TanyaI appreciate the diverse examples of how regular polygons are used in everyday life and various professional fields, showcasing their widespread relevance and application.

Teagan PalmerI appreciate the in-depth breakdown of the interior angle formula particularly. It adds a valuable mathematical perspective to the concept.

Ava26The relationship between regular polygons and computer graphics is fascinating, highlighting their role in creating visual representations and complex structures.

ZjonesThe comprehensive explanations of regular polygon properties and applications provided here offer valuable insights for those engaging with mathematical and geometric concepts.

Hall StephenThe connections between regular polygons and real-world applications in manufacturing, art, and navigation are clearly presented, highlighting their versatile nature.

Clark DaveThe relevance of regular polygons in navigation and surveying also stood out to me. It shows their practical importance beyond theoretical geometry.

Holmes ChristopherThe section on architecture and construction really emphasizes the visual and structural impact of regular polygons in designs and layouts.

Green SoniaI find the practical uses section particularly interesting as it demonstrates the real-world relevance of regular polygons.

Owen RobertsonThe real-world examples of how regular polygons are used in manufacturing and art are particularly interesting and really illustrate their significance.

Williams ErinThe connection to mathematical theorems and proofs is quite intriguing and adds depth to the study of regular polygons.

Kieran02Absolutely, the applications in fields like architecture, art, and navigation really showcase the versatility of regular polygons.