# The Regular Polygons

MATHEMATICS

### The Regular Polygons

### Regular polygons are fascinating geometric shapes that have a unique beauty and symmetry. They are defined as polygons where all sides are of equal length and all angles between the sides are of equal measure. Regular polygons have been studied for centuries by mathematicians and have many interesting properties that make them an important part of geometry.

The most well-known regular polygons are the triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, and decagon. Each of these polygons has a unique number of sides, ranging from three to ten. These shapes can be found in many natural and man-made objects, including crystals, flowers, and architecture.

One of the most interesting properties of regular polygons is that their interior angles are always multiples of 180 degrees divided by the number of sides. For example, a regular triangle has three sides, so each interior angle is 60 degrees (180/3 = 60). Similarly, a regular hexagon has six sides, so each interior angle is 120 degrees (180/6 = 120). This property makes it easy to calculate the interior angles of any regular polygon.

Another interesting property of regular polygons is that they can be inscribed in a circle. In other words, all of the vertices of the polygon lie on the circumference of the circle. The radius of the circle can be calculated using the formula R = s/2sin(180/n), where s is the length of one side of the polygon and n is the number of sides. This formula can be used to find the radius of any regular polygon that is inscribed in a circle.

Regular polygons also have a unique property known as rotational symmetry. This means that if you rotate the polygon around its center by a certain angle, it will look exactly the same as it did before the rotation. The angle of rotation that preserves the symmetry of the polygon is equal to 360 degrees divided by the number of sides. For example, a regular pentagon has five sides, so it has rotational symmetry of 72 degrees (360/5 = 72).

In addition to their mathematical properties, regular polygons are also aesthetically pleasing. The symmetry and simplicity of their shapes make them popular in art and design. For example, many logos and emblems use regular polygons as a basis for their designs.

In conclusion, regular polygons are fascinating geometric shapes that have many interesting properties. They are defined by their equal sides and angles, and have rotational symmetry and the ability to be inscribed in a circle. Regular polygons can be found in many natural and man-made objects, and are an important part of geometry and mathematics.

### Polygon

**Triangle**

**Square (or Quadrilateral)**

**Pentagon**

**Hexagon**

**Heptagon**

**Octagon**

**Nonagon**

**Decagon**

**Undecagon**

**Duodecagon (or Dodecagon)**

**Quindecagon**

**Icosagon**

**Polygon**

**3**

**4**

**5**

**6**

**7**

**8**

**9**

**10**

**11**

**12**

**15**

**20**

**60 degree**

**90 degree**

**108 degree**

**120 degree**

**128.57 degree**

**135 degree**

**140 degree**

**144 degree**

**147.27 degree**

**150 degree**

**156 degree**

**162 degree**

**Number of **

**sides**

**Internal angle**

** (Each)**

**Sum of internal**

** angles**

**180 degree**

**360 degree**

**540 degree**

**720 degree**

**900 degree**

**1080 degree**

**1260 degree**

**1440 degree**

**1620 degree**

**1800 degree**

**2340 degree**

**3240 degree**