## regular polygon

### A regular polygon is a polygon that has all sides of equal length and all angles of equal measure

A regular polygon is a polygon that has all sides of equal length and all angles of equal measure. In other words, it is a polygon that is both equilateral and equiangular. Regular polygons can have different numbers of sides, ranging from three (a regular triangle) to infinity.

Some common examples of regular polygons include:

1. Equilateral Triangle: A three-sided polygon with all sides and angles equal.

2. Square: A four-sided polygon with all sides and angles equal.

3. Pentagon: A five-sided polygon with all sides and angles equal.

4. Hexagon: A six-sided polygon with all sides and angles equal.

5. Octagon: An eight-sided polygon with all sides and angles equal.

The measure of each interior angle of a regular polygon can be found using the formula:

Interior Angle = (n-2) * 180° / n

where n is the number of sides of the polygon.

Similarly, the measure of each exterior angle of a regular polygon can be found using the formula:

Exterior Angle = 360° / n

Regular polygons have several interesting properties, including:

1. Symmetry: Regular polygons have multiple lines of symmetry passing through their center. These lines divide the polygon into congruent parts.

2. Diagonals: Regular polygons have diagonals, which are line segments connecting non-adjacent vertices. The number of diagonals in a regular polygon can be found using the formula:

Number of Diagonals = n * (n-3) / 2

3. Perimeter: The perimeter of a regular polygon can be calculated by multiplying the length of one side by the number of sides.

4. Apothem: The apothem of a regular polygon is the perpendicular distance from the center to any side. It bisects each side, and the ratio between the apothem and the side length is constant for all regular polygons.

Regular polygons have numerous applications in various fields, including architecture, design, and mathematics. Understanding their properties plays a crucial role in solving geometric problems and analyzing patterns in nature and man-made objects.

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