# Regular Polygons

If a polygon has:

*All sides of equal length,
*All angles of equal measure

then it is called a regular polygon. If not, it is irregular polygon.

In all n-gons, there is only one regular polygon, and its general shape is well-known to everyone; the others are irregular.
For example, in triangles, the equilateral triangle is regular, while the others are irregular.
In quadrilaterals, the square is regular, while the others are irregular.
In pentagons, the shape we call a regular pentagon is regular, while the others are irregular, and so on.

### Decagon (10 sides): Regular decagon

For regular polygons with more than ten sides, the names are often constructed using numerical prefixes combined with "-gon."

For example:

Dodecagon (12 sides): Regular dodecagon
Icosagon (20 sides): Regular icosagon

# Finding the Interior Angles of Regular Polygons

### Before we begin ;

What is the interior angle of a polygon?
-- It's the angle you find on the inside of each corner.

What is the exterior angle of a polygon?
-- It's the angle on the outside of the polygon, supplementary to the interior angle.

### Basic Rule:

*To find the properties of unknown shapes, utilize the characteristics of known shapes.
*The sum of the interior angles of a triangle is 180 degrees.

## Triangle - Equilateral Triangle

$$\displaystyle \frac{180°}{3} = 60°$$

*To find the properties of unknown shapes, utilize the characteristics of known shapes.

We know the sum of the interior angles of a triangle, and we can use this to find the sum of the interior angles of a square. Let's divide the square into two triangles by drawing a diagonal from one corner to another.

In this case, I have obtained two triangles, and if you look carefully, you will see that the sum of the interior angles of the two triangles is equal to the sum of the interior angles of the square.

a+b+c+d+e+f = Sum of the interior angles of the square

### a+b+c=180° d+e+f=180° a+b+c+d+e+f=360°

Sum of the interior angles of the square = 360°

To calculate one interior angle of the square, it's enough to divide by 4, because all four angles are equal to each other. (Regular polygon).

One interior angle of the square = $$\displaystyle \frac{360°}{4} = 90°$$

# Regular Pentagon

Let's apply the method we used for the square here as well. Draw the diagonals from one corner of the pentagon, and in this way, divide the pentagon into triangles.

I have obtained three triangles, and the sum of the interior angles of the three triangles equals the sum of the interior angles of the pentagon.

The sum of the interior angles of the three triangles is equal to the sum of the interior angles of the pentagon.

the sum of the interior angles of the pentagon 180° . 3 = 540°

To calculate one interior angle of the pentagon, it's enough to divide by 5, because all four angles are equal to each other. (Regular polygon).

One interior angle of the pentagon= $$\displaystyle \frac{5400°}{5} = 108°$$

interior angle of the regular pentagon

# Regular Hexagon

Let's take similar steps here and divide it into triangles.

the sum of the interior angles of the hexagon 180° . 4 = 720°

One interior angle of the hexagon= $$\displaystyle \frac{720°}{5} = 120°$$

# Regular Octagon

The sum of the interior angles of theOctagon 180° . 6 = 1080°

One interior angle of the Octagon= $$\displaystyle \frac{1080°}{8} = 135°$$

There's something you need to notice here: the number of triangles inside the polygons affects our angles. For example, simply put, as the number of sides of the polygon increases, the number of triangles inside it also increases, thereby increasing both the sum of the interior angles of the polygon and one of its interior angles. Now, let's see if there's a pattern between the number of sides of the polygon and the number of triangles.

# Exterior Angles of Polygons

There are two methods to find the exterior angles of polygons:
1) Calculate the interior angles, and since the interior and exterior angles are supplementary, subtract them from 180 degrees.

For example, if one interior angle of an octagon is 135 degrees, subtracting 135 degrees from 180 degrees gives an exterior angle of 45 degrees. Conversely, in an octagon, there are 8 exterior angles, and 8 * 45 = 360 degrees, confirming that our calculation is correct.

2) The sum of all exterior angles of a polygon is 360 degrees; if you divide 360 degrees by the number of sides, you will have calculated one exterior angle.

In an octagon, there are 8 exterior angles, and since they are equal to each other, dividing 360 degrees by 8 gives an exterior angle of 45 degrees.

# Regular Polygons in Life

The classification of polygons into regular and irregular categories serves several practical and theoretical purposes in mathematics, geometry, and various applied fields.

### Simplification of Mathematical Properties:

Regular polygons have symmetrical properties that make calculations involving areas, perimeters, and angles more straightforward. The uniformity in side lengths and angles allows for the development of general formulas.

### Understanding Symmetry:

Regular polygons exhibit both rotational and reflective symmetry. Studying these symmetries helps in understanding more complex geometric concepts and patterns.
Architectural and Design Applications:

Regular polygons are often used in design and architecture due to their aesthetically pleasing symmetrical properties. For example, hexagonal tiles can be used to tessellate a floor without gaps.

### Scientific and Technological Applications:

Regular polygons are used in various scientific fields. For example, the regular hexagonal shape of honeycomb cells in bee hives has been studied for its efficiency in packing and strength.

### Teaching Fundamental Concepts:

The study of regular polygons helps in teaching fundamental geometric concepts such as congruence, similarity, and transformations. It lays the groundwork for more advanced geometric and algebraic studies.

### Basis for Irregular Polygons:

Understanding the properties of regular polygons can also aid in the study of irregular polygons. By recognizing how irregular polygons deviate from regular ones, students can develop a deeper understanding of geometric properties and relationships.

### Computer Graphics and Gaming:

Regular polygons are often used in computer graphics, 3D modeling, and gaming as building blocks to create more complex shapes and textures.

### Cultural and Historical Significance:

Regular polygons have been studied since ancient times and have significant cultural and historical meanings in different societies. For example, the regular pentagon is associated with the golden ratio, a mathematical constant that has fascinated mathematicians, artists, and architects for centuries.