# Regular polygon

A regular polygon is a flat, simple polygon with sides of equal length and angles of equal measure.

## Perimeter of a Regular Polygon

$$C=n \times a$$

where,

n— number of sides, angles;
a— length of a side of the regular polygon.

## Area of a Regular Polygon

Area of a regular polygon through side and apothem:

$$S=\frac{n \times a \times r}{2},$$

n— number of sides, angles;
a— length of a side of the regular polygon;

Area of a regular polygon through apothem:

$$S=n\times R^{2}\sin\left ( \frac{180^{\circ}}{n} \right ),$$

n— number of sides, angles;

Area of a regular polygon through circumradius:

$$S=\frac{n\times R^{2}\sin\left ( \frac{360^{\circ}}{n} \right )}{2},$$

n— number of sides, angles;

Area of a regular polygon through side length:

$$S=\frac{n\times a^{2}}{4\tan\left ( \frac{180^{\circ}}{n} \right )},$$

n— number of sides, angles;
a— length of a side of the regular polygon.

Area of a regular polygon through perimeter:

$$S=\frac{C \times r}{2},$$

C— perimeter of the regular polygon;

## Interior Angle

Interior angle is the angle between two adjacent sides of a regular polygon, inside the polygon.

n— number of sides, angles.

The sum of interior angles can be found with the formula:

$$s=(n-2)180^{\circ},$$

n— number of sides, angles.

## Number of Diagonals

\begin{align} N&=\frac{1}{2}n(n-3),\\ \\ n&>2\\ \end{align}

n— number of sides, angles.