# What is Polygon?

It can be described in the following points;

**Polygon**is made up of two words:**poly**, which means “many,” and**gon**, which means “thing” (which means sides). For a figure to be closed, at least three line segments must connect end to end.- Several straight lines bound by it.
- It is said to be a regular polygon when all its sides and angles are equal.
- In geometry, it is a closed two-dimensional shape that is flat or plane and has straight sides.
- It does not have sides that curve.
- Its sides are also called
**edges**. Their corners are the points where two sides meet.

## Definition of Polygon

It can be defined as;

## Types of Polygons

There are 4 main types which are as follows;

**Regular****Irregular****Convex****Concave**

**Regular Polygon**

It can be described in the following points;

- It
**equilateral**triangle, the**square**, the regular**pentagon**, the regular**hexagon**, and the regular**octagon**are all examples of it. - In a regular polygon, all of the angles on the inside are also the same size. The formula below can be used to figure out how big each angle inside a regular polygon is:
**Formula of Interior angle**= (n – 2) x 180° / n.**n is the number of the side**. - They are helpful in many areas of maths, such as
**geometry**and**trigonometry**, because they have unique qualities that make them stand out. They also show up a lot in nature, like in the**honeycomb**structure of**beehives**and the shapes of some**crystals**.

### Irregular Polygon

It can be described in the following points;

- It does not have equal sides and its angles are said to be
**irregular**. In other words, a polygon with**uneven sides**and angles is one with a range of side and angle lengths and measurements. It contrasts with a regular polygon with equal sides and tips. - Any number of sides, sizes, and forms are possible for
**irregular polygons**. Triangles, quadrilaterals, pentagons, hexagons, and other shapes without congruent sides or angles are irregular polygons. Calculating a polygon’s irregularity might make it more difficult to determine some parameters, including size or perimeter, as each side and angle may need to be measured or computed separately.

### Convex Polygon

It can be described in the following points;

- Any polygon whose inner angles are less than
**180 degrees**and whose vertices face outward is considered**convex**. Its vertices do not protrude inward and its sides are not “**caved**in” or “**concave**.” - In contrast, a
**concave polygon**(also known as a non-convex) has at least one vertex pointing inward and at least one interior angle greater than 180 degrees. Because non-convex can contain “dents” or “hollows” in their shape, working with them mathematically can be more challenging. - It can be
**processed**and**analyzed**quickly, making them a popular subject for**computer science**and geometry study. They also have many valuable properties in mathematics. They are also frequently used in real-world settings, including computer graphics, design, and architecture.

### Concave polygon

Any polygon with at least one internal angle **greater than 180 degrees**—also called a **reflex angle**—is **concave**. In other terms, its boundary features a “**cave**” or indentation. In contrast, a convex polygon has no indentations on its edge, and all inside angles are smaller than 180 degrees.

## What is the Area of a Polygon?

It is the space that it takes up. It can be either regular or not. The most common polygons in geometry are **triangles**, **squares**, **rectangles**, **pentagons**, and **hexagons**. Each of these shapes has its area. For **regular polygons**, it’s easy to figure out how big they are because we already know their sizes. **For example**, the site of a square is easy to figure out if we see the length of one of its sides. This is because all of its sides are the **same length**.

You already know what the word “**area**” means. It is the area inside the edges of an object or figure that is flat. **m ^{2}** is a unit of area. There are formulas for squares, rectangles, circles, triangles, trapezoids, etc., that can be used to figure out the area of a shape. Conversely, the site is found by combining it with two or more regular polygons for an irregular polygon.

### Definition of Area of Polygon

### Inscribed Polygon

If a circle passes through the corners of a polygon, then this is called an **Inscribed polygon**, and the circle is called the **Circumscribed circle**. If it is regular in shape, the center of the circle is also the center of the polygon. In the given figure, “**O**” is the center, and “OF” is the radius of the circumscribed circle and is denoted by “**R**.”

### Circumscribed Polygon

It is the one in which the sides touch the circle **circumferences**. In this case, a circle is called an** Inscribed circle**. If it is regular, then the center of the circle is also the center of the polygon. The radius of the inscribed circle is denoted by “**r**” in the figure **OG **= **r**.

### Interior Angle

**The interior angle having n sides = (2n – 4/n) * 90 ^{o}.**

## Method for Finding Area of Regular Polygon

A regular polygon can be divided into equal isosceles triangles by joining all the corners to the center. The number of the triangle is the same as the number of sides of the polygon, then the area is equal to the size of any one triangle multiplied by the number of the triangle.

### Area of the regular polygon of “n” sides when the length of a side is given

Let AOB be one of the “n” triangles.

Then;

**<AOB = 360 ^{o} /n **

**and**

**<AOG = 180 ^{o} / n**

**Area = n * area of AOB**

**= n (AB * OG /2)**

**= n*(a/2)*OG**

Now, in the right triangle AOG;

**OG/AG = Cot (180 ^{o} / n)**

OG = AG **Cot (180 ^{o} / n)**

= a/2 **Cot (180 ^{o} / n)**

**Area ** = n* (a/2) * (a/2) **Cot (180 ^{o} / n)**

**Area ** =(na^{2} /4) **Cot (180 ^{o} / n)** Sq. Unit

**Perimeter = na**

### Area of a regular polygon of “n” sides when the radius of the inscribed circle “r” is given

**Area = n * area of AOB**

**= n (AB * OG /2)**

Now,

AB =?

OG = r

**AB = 2AG**

**AG/OG = tan (180 ^{o} / n)**

AG = OG **tan (180 ^{o} / n)**

AB / 2 = r **tan (180 ^{o} / n)**

AB = 2r **tan (180 ^{o} / n)**

**Area = n*AB (OG/2)**

**Area **= n*2r **tan (180 ^{o} / n)** *

**(r/2)**

**Area **=nr^{2} **tan (180 ^{o} / n)** Sq. Unit

**Perimeter **= 2nr **tan (180 ^{o} / n)**

### Area of a regular polygon of “n” sides when the radius of the circumscribed circle R is given

Let OA = R is the radius of the circumscribed circle

Area of Polygon = **n * area of AOB**

**= n* (OA * OB /2) Sin (360 ^{o} / n) **

**= n* (R* R/2) Sin (360 ^{o} / n)**

**Area of Polygon** = **(nR ^{2} /2)**

**Sin (360**Sq. Unit

^{o}/ n)## Difference b/w Area of & Perimeter of Polygon

The perimeter and area are both numbers that can be measured depending on how long each side is. To be able to tell them apart, you need to know what the primary difference is between perimeter and area.

Criteria of Difference | Perimeter of Polygon | Area of Polygon |
---|---|---|

Definition | The total length of the polygon’s border is found by adding up the lengths of all its sides. | It is the area or space that is surrounded by a polygon. |

Formula | The perimeter of the Polygon = Length of Side 1 + Length of Side 2 + …+ Length of side N (if the number of sides is N) | Different formulas can be used to find the area of a polygon based on whether it is a regular polygon or an irregular polygon. |

Unit | Perimeters are measured in meters, centimeters, inches, feet, etc. | The unit of the area of polygons is (m)^{2}, (cm)^{2}, (inch)^{2}, (ft)^{2}, etc. |

## Area of Polygon Formulas

Depending on the sides’ length, it can be either regular or irregular. So, this difference also makes a difference in how the area of polygons is calculated.

Name | Area Formula |

Area of Triangle | 1/2 × base × height |

Area of Square | side^{2} |

Area of Rectangle | length × width |

Area of Pentagon | 5/2 × side length × distance from the center of sides to the center of the pentagon |

Area of Rhombus | 1/2 × product of diagonals |

Area of Hexagon | (3√3)/2 × distance from the center of sides to the center of the hexagon |

## FAQ’s

### What type of polygon is a star?

Non-convex polygon A star shape is a type of non-convex polygon in geometry. This is a shape in which no angles are more than 180 degrees. Most of the time, a star shape is also a decagon.

### What is a 20 polygon called?

In geometry, a 20-gon or icosagon is a polygon with twenty sides. The sum of the angles on the inside of an icosagon is 3240 degrees.

### Who created the polygon?

India made Polygon in 2017. It was first called the Matic Network. It was the idea of Jayanti Kanani, Sandeep Narwal, Anurag Arjun, and Mihailo Bjelic, all of whom had worked on Ethereum before.

### What are the 5 properties of a polygon?

They have certain traits. Two-dimensional shapes can be changed in simple ways. Circles and other conditions with curves. Things about polygons How many sides the body has? The shape’s sides make the angles. How long each side of the figure is.

### What is the polygon law?

The polygon law of vector addition says that if you can show the size and direction of several vectors by taking the sides in the same order, then you can show the size and direction of their sum by taking the last side of the polygon in the opposite order.

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