# Regular Polygon

## Understanding Regular Polygons: Their Definition, Properties, and Examples

A regular polygon is a closed shape with straight edges and equal sides and angles. Based on the size of its interior angles, it is classified as either convex or concave. A convex polygon has interior angles less than 180° each, while a concave polygon has at least one interior angle greater than 180°. Regular polygons are primarily convex and possess unique properties that will be explored in this article.

### Defining Regular Polygons

A regular polygon has equal side lengths and equal interior angles. Examples of regular polygons include equilateral triangles, squares, and rhombuses. In addition, a regular polygon's diagonals are all of equal length. While most regular polygons are convex, there are also some concave regular polygons that have a star shape. Now, we will take a closer look at the properties of regular convex polygons.

### Properties of Regular Convex Polygons

**Two Important Circles:**A regular convex polygon has two significant circles associated with it. The circumcircle lies outside the polygon and passes through all of its vertices, while the incircle passes through the midpoint of each of its sides. The circumcircle's radius is the distance from the polygon's center to any of its vertices, and the incircle's radius is the distance between the center and the midpoint of any side, also known as the apothem.**Calculating Area:**One interesting application of a regular polygon's properties is the ability to estimate its area using the apothem. By breaking the polygon down into triangles and utilizing the apothem, we can determine the area of any regular polygon with N sides.

### An Example Calculation:

To calculate the area of a regular hexagon with side length s and apothem l, we can divide it into six triangles. The area of one triangle is equal to 0.5 x s x l, and to find the area of the entire hexagon we multiply it by 6 (the number of sides). Therefore, the formula for the area of a regular hexagon is 6 x 0.5 x s x l, or 3 x s x l.

### Examples of Regular Polygons

A regular polygon with three sides is known as an equilateral triangle, and one with four sides is called a square. For polygons with more than four sides, the term "regular" is added before the name of the polygon, such as a regular pentagon. Some examples of regular (equiangular) convex polygons include regular hexagons, octagons, and decagons.

### Formulas for Regular Polygons

**Exterior Angles:**In a regular convex polygon, the sum of all exterior angles is always 360° or 360/N, where N is the number of sides and ∅ is the exterior angle.**Interior Angles:**The sum of the interior angles in a regular polygon depends on the number of sides it has. For example, a triangle has a sum of 180°, while a quadrilateral has a sum of 360°. A general equation for finding the sum of interior angles in any regular polygon is 180(N-2), where N is the number of sides.

Regular polygons possess unique properties and formulas related to their side lengths, angles, and area. Understanding these properties can assist in solving various problems involving regular polygons.

## Understanding Interior and Exterior Angles in Polygons

In a polygon, the interior angle is the angle formed by two consecutive sides on the inside of the figure. On the other hand, the exterior angle is the angle formed by one side and the extension of the adjacent side. Let's examine how we can find these angles for a regular decagon.

**Interior Angle:** To determine the interior angle of a regular decagon, we can use the formula 180° - 360°/N, where N is the number of sides. For a decagon, N=10, so the interior angle would be 180° - (360°/10) = 144°. This means that each interior angle in a regular decagon measures 144°.

**Exterior Angle:** To find the exterior angle of a regular decagon, we can divide 360° by the number of sides, or N. For a decagon, this would be 360°/10 = 36°.

**Sum of Interior Angles:** The sum of all interior angles in a regular polygon can be calculated using the formula (N-2) x 180°, where N is the number of sides. For a regular decagon, this would be (10-2) x 180° = 1440°.

## The Beauty of Convex Polygons: Understanding Diagonals

A convex polygon is defined as a shape where all its interior angles are less than 180 degrees, and its edges never cross. Unlike concave polygons, where diagonals can extend to the exterior of the shape, the diagonals in a convex polygon will always stay within the figure, adding to its visual appeal and symmetry. A diagonal is a line segment connecting any two non-consecutive vertices of a polygon with more than 3 sides. The number of diagonals in a convex polygon with 'N' sides can be easily calculated using the formula N(N-3)/2.

For example, a heptagon, a convex polygon with 7 sides, has 14 diagonals. This can be obtained by substituting N=7 in the formula, giving us N(N-3)/2 = 7(7-3)/2 = 14 diagonals. In the figure shown, all 14 diagonals of a regular heptagon can be seen.

## The Special Properties of Regular Polygons

A regular polygon is a shape with all its sides and interior angles being equal. It exhibits unique properties that make it stand out among other polygons. Some of these properties include:

- All sides and interior angles are equal.
- The diagonals are also equal in length.
- The circumcircle, a circle passing through all the vertices, has a radius known as the circumradius.
- The incircle, a circle passing through the midpoints of each side, has a radius called the apothem.
- The area of a regular polygon can be calculated by dividing it into smaller triangles.
- All the vertices are equidistant from the center of the polygon.
- The sum of all exterior angles is always equal to 360°.

## Frequently Asked Questions

**What Defines a Regular Polygon?** A regular polygon is a shape with equal sides and interior angles, creating a perfect symmetry.

**What is the Minimum Number of Sides a Regular Polygon Can Have?** A regular polygon must have at least 3 sides, but it can have an unlimited number of sides.

**What are Some Examples of Regular Polygons?** Regular polygons come in various shapes, including equilateral triangles, squares, and rhombuses.

**How Do You Find the Area of a Regular Polygon?** The area of a regular polygon can be calculated by dividing it into triangles and adding up their individual areas.

**What Shapes Do Regular Polygons Have?** Regular polygons can take on various shapes depending on the number of sides, but all sides and angles are equal, creating a perfect symmetry.