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# Polygon calculator - diagonals, area, perimeter, sides lenght

Regular polygon calculator will help you calculate the diagonals of any regular polygon, side length, height, area, perimeter and radius of the circumscribed circle and the radius of the circle inscribed in a regular polygon.

## Perimeter of a regular polygon

$$L=a \cdot n$$

## Height of a regular polygon

$$h=\frac{2\cdot a}{2\cdot \tan(\frac{\pi}{n})} \hspace{1mm}for \hspace{1mm}n \hspace{1mm}even$$ $$h=\frac{a}{2\cdot \tan(\frac{\pi}{\frac{2}{n}})} \hspace{1mm}for \hspace{1mm}n \hspace{1mm}odd$$

## Regular polygon area

$$S=\frac {1}{4}\cdot n \cdot a^{2}\cdot \cot(\frac {\pi }{n}) = \frac{n\cdot a^{2}}{4\cdot \tan(\frac{\pi}{n})}$$

## Radius of a circle circumscribed on a regular polygon

$$R=\frac {a}{2\cdot \sin(\frac{\pi }{n})}$$

## Radius of the circle inscribed in a regular polygon

$$r=\frac {a}{2\cdot \tan(\frac {\pi }{n})} = \frac {a}{2}\cdot \cot(\frac {\pi }{n})$$

## Number of diagonals of a regular polygon

$$d=\frac {n(n-3)}{2}$$

## Diagonal lengths of a regular polygon

$$d_k=\frac{a\sin\frac{(k+1)\pi}{n}}{\sin\frac{\pi}{n}},$$ where $$k\in\mathbb{N},\ 1\le k\le n-3\,$$

## Inside angle measure and a measure of the central angle of a regular polygon

Inside angle measure (between adjacent sides): $$\gamma =\frac{\pi (n-2)}{n}\mathrm{rad} =\frac{180^{\circ }\cdot (n-2)}{n}$$ Measure of the center angle (that is, the angle at which the side of the polygon is viewed from its center): $$\beta =\frac {2\pi }{n}\mathrm {rad} =\frac {360^{\circ }}{n}$$

Regular polygon - information

Regular polygon - a polygon that has all interior angles equal and all sides of equal length. The smallest possible number of sides of a regular polygon is 3. Theoretically it is possible to construct a regular diagonal, but this is a degenerate case, it would look like a regular segment and the angle between the sides would be 0 °.

It has the following properties:
• a – length of one side of the polygon;
• n – lthe number of sides of a regular polygon, where $$n\in\mathbb{N}, n > 2.$$
1. Formula for the perimeter of a regular polygon: $$L=n \cdot a$$
2. Formulas for the height of a regular polygon: $$h=\frac{2\cdot a}{2\cdot \tan(\frac{\pi}{n})} \hspace{1mm}for \hspace{1mm}n \hspace{1mm}even$$ $$h=\frac{a}{2\cdot \tan(\frac{\pi}{\frac{2}{n}})} \hspace{1mm}for \hspace{1mm}n \hspace{1mm}odd$$
3. Formulas for the area of a regular polygon: $$S=\frac{1}{4}na^2\operatorname{ctg}\frac{\pi}{n}$$ $$=\frac{nar}{2}$$ $$=nr^2\operatorname{tg}\frac{\pi}{n}$$ $$=nR^2\sin\frac{\pi}{n}\cos\frac{\pi}{n}$$ $$=\frac{1}{2}nR^2\sin\frac{2\pi}{n}$$
4. Formula for the radius of a circle described on a regular polygon: $$R=\frac{a}{2\sin\frac{\pi}{n}}=\frac{a}{2}\operatorname{csc}\frac{\pi}{n}$$
5. Formula for the radius of a circle inscribed in a regular polygon: $$r=\frac{a}{2\operatorname{tg}\frac{\pi}{n}}=\frac{a}{2}\operatorname{ctg}\frac{\pi}{n}$$
6. Formula for the number of diagonals of a regular polygon: $$d=\frac {n(n-3)}{2}$$
7. Formula for the length of the diagonals of a regular polygon: $$d_k=\frac{a\sin\frac{(k+1)\pi}{n}}{\sin\frac{\pi}{n}},$$ where $$k\in\mathbb{N},\ 1\le k\le n-3\,$$
8. Formula for the side length of a regular polygon: $$a=2\sqrt{R^2-r^2}$$ $$=2R\sin {\frac {\pi }{n}}$$ $$=2r\operatorname {tg} {\frac {\pi }{n}}$$
9. The angle between any adjacent diagonals originating from one vertex (including sides originating from that vertex) $$\gamma ={\frac {\pi }{n}}\mathrm {rad} ={\frac {180^{\circ }}{n}}$$
10. Formula for the measure of the between adjacent sides of a regular polygon: $$\gamma =\frac{\pi (n-2)}{n}\mathrm{rad} =\frac{180^{\circ }\cdot (n-2)}{n}$$
11. Formula for the center angle (the angle at which the side of the polygon is seen from its center): $$\beta =\frac {2\pi }{n}\mathrm {rad} =\frac {360^{\circ }}{n}$$

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