**Regular Polygon: A Comprehensive Guide**

A regular polygon is a **polygon with n sides**, characterized by equal side lengths and symmetrical placement around a central point. Notably, only specific regular polygons can be constructed using classical Greek tools like the compass and straightedge.

The terms “equilateral triangle” and “square” precisely describe regular polygons with 3 and 4 sides, respectively. For polygons with n>=5 sides, such as pentagon, hexagon, and so on, these terms may refer to either regular or irregular polygons. However, in most cases, they are assumed to refer to regular polygons unless specified otherwise.

**Definition of a Regular Polygon**

A regular polygon is a polygon that has all sides of equal length and all angles of equal measure. This uniformity in sides and angles gives regular polygons a distinct symmetry that sets them apart from irregular polygons.

** Characteristics of Regular Polygons**

**Equal Sides:**In a regular polygon, all sides have the same length, denoted as “s.”**Equal Angles:**All interior angles of a regular polygon are congruent, meaning they have the same degree measure.**Center:**Regular polygons have a center point equidistant from all vertices, known as the circumcenter.**Symmetry:**Regular polygons exhibit rotational symmetry, where you can rotate the polygon around its center by a certain degree to align with its original position.

**Types of Regular Polygons**

Regular polygons are named based on the number of sides they possess:

- Three sides: Equilateral Triangle
- Four sides: Square
- Five sides: Pentagon
- Six sides: Hexagon
- Seven sides: Heptagon
- Eight sides: Octagon
- Nine sides: Nonagon
- Ten sides: Decagon

**Formulas for Regular Polygons**

**1. Interior Angle**

The formula to calculate the interior angle of a regular polygon is:

Interior Angle (in degrees) = (n – 2) * 180° / n, where “n” is the number of sides.

**2. Exterior Angle**

The exterior angle of a regular polygon can be found by subtracting the interior angle from 180 degrees.

**3. Perimeter of a Regular Polygon
**

The perimeter of a regular polygon is calculated by multiplying the number of sides (n) by the length of one side (s).

**4. Apothem**

The apothem is the distance from the center of a regular polygon to the midpoint of a side.

Its formula is:

Apothem = s / (2 * tan(π/n)), where “n” is the number of sides.

## Solved Examples on Regular Polygon

###### Example 1: Find the interior angle of a regular hexagon.

**Solution:** A regular hexagon has six sides. To find the interior angle, we can use the formula for the interior angle of a regular polygon:

Interior Angle (in degrees) = (n – 2) * 180° / n

In this case, n = 6 (since it’s a hexagon):

Interior Angle = (6 – 2) * 180° / 6 Interior Angle = (4) * 180° / 6 Interior Angle = 720° / 6 Interior Angle = 120°

So, the interior angle of a regular hexagon is 120 degrees.

**Example 2: Calculate the apothem of a regular octagon with a side length of 8 cm.**

**Solution:** An apothem is the distance from the center of a regular polygon to the midpoint of a side. To find the apothem, we can use the formula:

Apothem = Side Length / (2 * tan(π/n))

In this case, n = 8 (since it’s an octagon) and the side length is 8 cm:

Apothem = 8 cm / (2 * tan(π/8)) Apothem = 8 cm / (2 * tan(π/8)) Apothem ≈ 4 cm (rounded to the nearest centimeter)

So, the apothem of the regular octagon is approximately 4 cm.

##### Example 3: You have a regular pentagon with an interior angle of 108 degrees. Find the exterior angle.

**Solution:** The exterior angle of a regular polygon is found by subtracting the interior angle from 180 degrees.

Exterior Angle = 180° – Interior Angle

Angle = 180° – 108°

Exterior Angle = 72°

So, the exterior angle of the regular pentagon is 72 degrees.

## Different Regular Polygons

Regular Polygon | Number of Sides | Characteristics |
---|---|---|

Equilateral Triangle | 3 sides | – All sides are of equal length. – Each interior angle measures 60 degrees. – Sum of interior angles: 180 degrees. |

Square | 4 sides | – All sides are of equal length. – Each interior angle measures 90 degrees. – Sum of interior angles: 360 degrees. |

Pentagon | 5 sides | – All sides are of equal length. – Each interior angle measures 108 degrees. – Sum of interior angles: 540 degrees. |

Hexagon | 6 sides | – All sides are of equal length. – Each interior angle measures 120 degrees. – Sum of interior angles: 720 degrees. |

Heptagon (Septagon) | 7 sides | – All sides are of equal length. – Each interior angle measures approximately 128.57 degrees. – Sum of interior angles: 900 degrees. |

Octagon | 8 sides | – All sides are of equal length. – Each interior angle measures 135 degrees. – Sum of interior angles: 1080 degrees. |

Nonagon (Enneagon) | 9 sides | – All sides are of equal length. – Each interior angle measures approximately 140 degrees. – Sum of interior angles: 1260 degrees. |

Decagon | 10 sides | – All sides are of equal length. – Each interior angle measures 144 degrees. – Sum of interior angles: 1440 degrees. |

## FAQs

**1. What is a regular polygon?**

A regular polygon is a two-dimensional geometric shape with equal-length sides and equal interior angles.

**2. How do you determine if a polygon is regular?**

To be considered regular, a polygon must have all its sides of equal length and all its interior angles of equal measure.

**3. What are some examples of regular polygons?**

Common examples include the equilateral triangle (3 sides), square (4 sides), pentagon (5 sides), hexagon (6 sides), heptagon (7 sides), octagon (8 sides), nonagon (9 sides), and decagon (10 sides).

**4. What is the formula to calculate the interior angle of a regular polygon?**

The formula is: Interior Angle (in degrees) = (n – 2) * 180° / n, where “n” is the number of sides.

**5. How do you calculate the exterior angle of a regular polygon?**

The exterior angle is found by subtracting the interior angle from 180 degrees.

**6. Can regular polygons have curved sides?**

No, regular polygons have straight sides. If a polygon has curved sides, it is not considered regular.

**7. What is the significance of the circumcenter in regular polygons?**

The circumcenter is the point equidistant from all vertices of a regular polygon. It plays a crucial role in determining the symmetry and properties of the polygon.

**8. Are there regular polygons with more than 10 sides?**

Yes, regular polygons can have any number of sides, but their names are less commonly used for polygons with sides beyond decagon (10 sides).

**9. Where are regular polygons commonly encountered in real life?**

Regular polygons are found in various architectural designs, such as the square and octagon in building structures, and in everyday objects like stop signs, which are octagonal.

**10. How is the apothem of a regular polygon calculated?**

The apothem is found using the formula: Apothem = Side Length / (2 * tan(π/n)), where “n” is the number of sides.