Decagons, or 10-sided polygons, are fascinating shapes that have intrigued mathematicians and scientists for centuries.

If you’re short on time, here’s a quick answer to your question:** a decagon has 10 sides.**

In this article, we’ll take a closer look at decagons and explore their properties, formulas for calculating their area and perimeter, and real-life applications.

## What is a decagon?

A decagon is a polygon with 10 sides and 10 angles. The word “decagon” comes from the Greek words “deka” meaning “ten” and “gonia” meaning “angle”. Decagons can come in many different shapes and sizes, but all have 10 sides.

### Types of decagons

There are two types of decagons: regular and irregular decagons.

- A regular decagon has equal sides and equal angles. Each interior angle of a regular decagon measures 144 degrees.
- An irregular decagon has sides and angles of different lengths and sizes.

### Properties of decagons

Decagons have several interesting properties:

Property | Regular decagon | Irregular decagon |
---|---|---|

Number of sides | 10 | 10 |

Number of angles | 10 | 10 |

Sum of interior angles | 1440 degrees | Varies |

Sum of exterior angles | 360 degrees | Varies |

Area | (1/4) × (5 + √5) × a^{2}, where a is the length of a side |
Varies |

Perimeter | 10a, where a is the length of a side | Varies |

Decagons can be found in many different places in nature and in architecture. For example, the United States Department of the Treasury uses a decagon-shaped building for their headquarters in Washington, D.C.

## Formulas for calculating the area and perimeter of a decagon

Decagons are geometric shapes that have ten sides and ten angles. They can be regular or irregular, depending on the length of their sides and angles. Here are some formulas for calculating the area and perimeter of both regular and irregular decagons:

### Area of a regular decagon

A regular decagon has ten equal sides and ten equal angles. To calculate the area of a regular decagon, you can use the following formula:

**Area = (5/4) x s^2 x √(5 + 2√5)**

Where s is the length of one of the sides of the decagon.

This formula may seem complicated, but it is derived from a simpler formula for the area of a regular polygon, which is:

**Area = (1/2) x apothem x perimeter**

Where apothem is the distance from the center of the polygon to the midpoint of one of its sides. For a regular decagon, the apothem is given by:

**Apothem = (s/2) x √(5 – 2√5)**

By substituting this value for apothem in the formula for the area of a regular polygon, we can derive the formula for the area of a regular decagon given above.

### Perimeter of a regular decagon

The perimeter of a regular decagon is simply the sum of the lengths of its ten sides. Since all sides of a regular decagon are equal, we can calculate the perimeter by multiplying the length of one side by ten:

**Perimeter = 10 x s**

### How to calculate the area and perimeter of an irregular decagon

Unlike regular decagons, irregular decagons have sides and angles of different lengths and sizes. As a result, there is no single formula for calculating their area and perimeter. However, we can still calculate their area and perimeter by dividing them into smaller regular polygons and adding up their areas and perimeters.

For example, we can divide an irregular decagon into five triangles, each with a common vertex at the center of the decagon. By calculating the area and perimeter of each triangle using the formulas for regular triangles, we can then add up their areas and perimeters to get the area and perimeter of the irregular decagon.

Alternatively, we can divide an irregular decagon into a combination of triangles and trapezoids, and then calculate the area and perimeter of each shape using the appropriate formulas for regular polygons.

## Real-life applications of decagons

Decagons, polygons with ten sides, have numerous real-life applications. They can be found in various fields such as architecture, art, and science.

### Architecture

Decagons are used in architecture for their unique shape and symmetry. For example, the famous US Pentagon building is a decagon-shaped structure. The design of the building is inspired by classical architecture, where the use of geometric shapes was common. The decagon shape provides a great balance between symmetry and functionality, making it a popular choice for architects.

Moreover, decagons can be used in the construction of domes and cupolas. The shape of a decagon allows for a more stable and sturdy structure as compared to other shapes like triangles or squares. This is because the angles of a decagon are equal, and the sides are long enough to provide support.

### Art

Decagons are also used in art for their unique aesthetic appeal. In fact, many artists use decagons in their work to create visually stunning pieces. The symmetry and balance of the shape make it an ideal choice for creating patterns and designs.

Additionally, decagons are used in various Islamic art forms like tiling and calligraphy. The geometric patterns created using decagons are not only aesthetically pleasing but also have spiritual significance in Islamic culture. These patterns are often used to represent the infinite nature of Allah.

### Science

Decagons can also be found in science, particularly in the study of crystals. In crystallography, decagons are used to describe the shape of certain crystals. For example, the crystal structure of the mineral pyrite is in the shape of a decagon.

Furthermore, decagons can be used in the study of viruses. The capsids of some viruses have a decagon shape, which provides structural stability and allows for efficient packaging of the virus genome.

## Conclusion

In conclusion, decagons are fascinating shapes that have both mathematical and real-life applications. They may seem simple at first, but they have many interesting properties and formulas that make them worth studying.

Whether you’re an artist, architect, scientist, or just someone who loves geometry, understanding decagons can open up a world of possibilities and inspire new ideas.