**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**

## Introduction

Regular polygons are a fundamental element of geometry, encountered in various real-world applications, from architecture and design to engineering and science. Calculating the properties of regular polygons can be a tedious task, especially when dealing with polygons with a high number of sides. To simplify these calculations, mathematicians and developers have created the “Regular Polygon Calculator,” a versatile tool that makes it easy to determine various attributes of regular polygons, such as their side length, perimeter, area, and interior angles.

## Concept of Regular Polygons

A regular polygon is a closed two-dimensional shape with equal side lengths and equal interior angles. The most well-known regular polygon is the equilateral triangle, with three equal sides and three equal angles of 60 degrees each. Other common regular polygons include squares (four sides, four right angles), pentagons (five sides), hexagons (six sides), and so on. The properties of a regular polygon depend on the number of sides it has, which is represented by the variable “n.”

## Formulae for Regular Polygons

### Interior Angle of a Regular Polygon (θ):

The interior angle of a regular polygon can be calculated using the following formula: θ = (180° * (n – 2)) / n

Where:

- θ is the interior angle in degrees.
- n is the number of sides of the polygon.

### Exterior Angle of a Regular Polygon (α):

The exterior angle of a regular polygon is complementary to the interior angle and can be calculated as: α = 360° / n

Where:

- α is the exterior angle in degrees.
- n is the number of sides of the polygon.

### Apothem (a):

The apothem is the distance from the center of a regular polygon to the midpoint of any side. It can be calculated using: a = s / (2 * tan(π / n))

Where:

- a is the apothem.
- s is the length of a side.
- n is the number of sides of the polygon.

### Perimeter (P):

The perimeter of a regular polygon is the total length of all its sides and can be calculated as: P = n * s

Where:

- P is the perimeter.
- s is the length of a side.
- n is the number of sides of the polygon.

### Area (A):

The area of a regular polygon can be calculated using the following formula: A = (1/2) * n * s * a

Where:

- A is the area.
- s is the length of a side.
- n is the number of sides of the polygon.
- a is the apothem.

## Example Calculations

Let’s calculate the interior angle, exterior angle, apothem, perimeter, and area of a regular hexagon (a polygon with six sides) with a side length of 4 units.

- Interior Angle (θ): θ = (180° * (6 – 2)) / 6 θ = (180° * 4) / 6 θ = 720° / 6 θ = 120°
- Exterior Angle (α): α = 360° / 6 α = 60°
- Apothem (a): a = 4 / (2 * tan(π / 6)) a ≈ 3.464 units (rounded to three decimal places)
- Perimeter (P): P = 6 * 4 P = 24 units
- Area (A): A = (1/2) * 6 * 4 * 3.464 A ≈ 41.568 square units (rounded to three decimal places)

## Real-World Use Cases

The Regular Polygon Calculator finds applications in various fields:

### Architecture and Design

Architects use regular polygons in designing buildings, floor plans, and facades. Calculating the dimensions and angles of regular polygons helps in creating aesthetically pleasing and structurally sound designs.

### Engineering

Engineers use regular polygons in various engineering disciplines, including civil, mechanical, and electrical engineering. For instance, in civil engineering, the calculator can be used to determine the dimensions of regular-shaped structures like bridges and tunnels.

### Computer Graphics

In computer graphics and game development, regular polygons play a crucial role in creating 3D models and animations. The calculator assists in defining and rendering polygons accurately.

### Chemistry

Chemists encounter regular polygons when studying molecular structures. Understanding the geometry of these structures is essential for predicting chemical properties and reactions.

### Surveying and Land Management

Land surveyors use regular polygons to divide land into plots or parcels. The Regular Polygon Calculator aids in determining the dimensions and angles needed for accurate land division.

## Conclusion

The Regular Polygon Calculator is a valuable tool that simplifies the calculation of various attributes of regular polygons. It provides a convenient way to determine interior angles, exterior angles, apothems, perimeters, and areas for polygons with any number of sides. This tool has widespread applications in architecture, engineering, computer graphics, chemistry, surveying, and other fields where regular polygons are encountered.

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