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;
r— apothem or inradius (radius of the inscribed circle).
Area of a regular polygon through apothem:
$$S=n\times R^{2}\sin\left ( \frac{180^{\circ}}{n} \right ),$$
n— number of sides, angles;
r— apothem or inradius (radius of the inscribed circle).
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;
R— circumradius or circumcircle radius (radius of the circumscribed circle).
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;
r— apothem or inradius (radius of the inscribed circle).
Interior Angle
Interior angle is the angle between two adjacent sides of a regular polygon, inside the polygon.
\begin{align} \alpha&=\frac{180^{\circ}(n-2)}{n},\\ \alpha&=\frac{\pi(n-2)}{n} \textrm{rad},\\ \end{align}
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.
See also:
Frequently asked questions
What is a regular polygon?
A regular polygon has all sides equal and all interior angles equal.
How do you calculate the perimeter of a regular polygon?
Multiply the number of sides by the length of one side.
How do you calculate the area of a regular polygon?
The area can be found using the side length and apothem, the circumradius, or the perimeter and apothem.
