Choose language

PL, EN, ES, DE, FR, RU


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): $$ \beta =\frac {2\pi }{n}\mathrm {rad} =\frac {360^{\circ }}{n} $$





Users of this calculator also used

Rail Fence, Zig-Zag - encoder / decoder

Rail Fence, Zig-Zag cipher online encoder and decoder. Encrypt and decrypt any cipher created in a Rail Fence, Zig-Zag cipher.

Kinetic energy calculator

With this calculator you can quickly calculate the kinetic energy of an object. You will also calculate mass from energy and velocity and speed given kinetic energy and mass.

Beaufort cipher - encoder / decoder

Beaufort cipher online encoder and decoder. Encrypt and decrypt any cipher created in a Beaufort cipher.

COTANGENT calculator

With the COTANGENT trigonometric function calculator you can calculate the values of any cotangent function. In addition to the response results, the calculator will also plot the selected function. You can choose one of the preset functions, e.g. cotangent, cot2 - cotangent square, arccot - arcus cotangent, coth - hyperbolic cotangent, arcoth - inverse function to coth, you can also enter your own function, e.g. cot(x) * cot(x) * cot(x) for cot3(x), cot(2x), cot(x+3), cot(x^2) etc.

Online tool for drawing graphs of any function.

With this online function graph plotter, you can draw any function. On one graph you can draw any three functions and compare their parameters. You can create graphs for many equations and functions.

Four-square cipher - encoder / decoder

Four-square cipher online encoder and decoder. Encrypt and decrypt any cipher created in a Four-square cipher. You can use any shift and additionally a key for more precise text encoding.

Average absolute deviation, or mean absolute deviation (MAD)

With this calculator you will calculate the average absolute deviation, or mean absolute deviation (MAD), of a data set is the average of the absolute deviations from a central point.



Online calculator