Supplementary mathematics/Regular polygon

"Regular polygon" in Euclidean geometry is a polygon whose angles and sides are equal. Regular polygons can be cuboid or star-shaped. In the limiting case, a sequence of regular polygons with increasing number of sides becomes a circle if the perimeter remains constant and becomes an apeirogon if the length of the side remains constant.

perimeter
The perimeter of a regular polygon is based on the multiplication of the number of sides of a regular polygon and the size of the sides of a regular polygon.

The perimeter of the area is obtained based on this relationship. $$P=a_1+a_2+...+a_n=(n.a)=na$$ Here n is equal to the number of sides of a regular polygon and a here is equal to the size of the sides of a regular polygon Is.

Area
The area of a regular polygon is obtained based on trigonometric relationships. The area of a regular polygon is based on the fact that it is made of 1x1 squares and the number of its sides is n, and because it is based on the number pi, the number of sides expands trigonometrically based on the cotangent in the form of pi. It is obtained by dividing the number of sides of a regular polygon.

The area of a regular polygon is written accordingly:

$$A=\frac{1}{4}n.a^2 \cot(\frac{\pi}{n})$$

Here, pi is in radians (equal to 180°).

The square area relationship by trigonometric method
Since the square is a regular polygon, its area can also be written as the area of a regular polygon which is obtained by trigonometric method, which is as follows: $$\frac{1}{4}na^2 \cot \frac{\pi}{n}=a^2 \cot \frac{\pi}{4}=a^2.(\sqrt{1})=1.a^2=a^2$$ here:

$$\frac{1}{4}na^2=\frac{1}{4}4a^2=a^2$$

$$\cot \frac{\pi}{4}=1$$

Therefore, the area of a parallel square is equal to the square of its side.

Other other formula area
The area of an "n-" regular polygon with the size of the side a, the radius of the surrounding circle "R", the radius of the surrounding circle "r" and the perimeter "p" is obtained using the following relations:

(The angles are in radians.)


 * $$A= \tfrac{1}{2}nar = \tfrac{1}{2}pr = \tfrac{1}{4}na^2\cot{\tfrac{\pi}{n}} = nr^2\tan{\tfrac{\pi}{n}} = \tfrac{1}{2}nR^2\sin{\tfrac{2\pi}{n}}$$

where R is equal to:


 * $$R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} = \frac{a}{\cos\left(\frac{\pi}{ n} \right)}$$