## Properties of an n-gon

This tool calculates the basic geometric properties of a regular convex n-gon, that is a polygon having n sides and n vertices. Regular polygons are equilateral (all sides equal). The tool can calculate the properties of any convex n-gon, given the length of its sides, or the inradius or the circumradius. Enter below the shape dimensions. The calculated results will have the same units as your input. Please use consistent units for any input.

Known data: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

N = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Geometric properties: | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Area = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Perimeter = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Side α = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Inradius R _{i} = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Circumradius R _{c} = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Angles : | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Interior φ = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

Central θ = |

## Definitions

### Geometry

N-gon is a polygon having N sides and N vertices. An n-gon is regular when all its sides are equal. The smallest regular n-gon can be considered the regular triangle, with three equal sides (equilateral triangle). The regular quadrilateral polygon is the square. The discussion in this page is about convex polygons (all interior angles are smaller than 180°). Regular polygons can also be non-convex (stars).

The interior angles of a regular pentagon are always same, given by the formula:

\[ \varphi = \pi-\frac{2\pi}{N} \]

where N the number of sides and vertices. For non-trivial cases it is assumed that \(N \ge 3\).

For any regular polygon, a circle that passes through all vertices can be drawn. That is the cirmuscribed circle or circumcircle of the polygon. Also, a circle that is tangent to all sides can be drawn, which is called inscribed circle or incircle. The radius of circumcircle R_{c} and incircle R_{i} (usually called circumradius and inradius respectively), are related to the side length α and also among each other. These relationships can be discovered using the properties of the right triangle, highlighted in the figure below, leading to the following formulas:

\[ \begin{split} R_c & = \frac{a}{2 \sin{\frac{\theta}{2}}} \\ R_i & = \frac{a}{2 \tan{\frac{\theta}{2}}} \\ R_i & = R_c \cos{\frac{\theta}{2}} \end{split} \]

where θ the central angle and α the side length. For a regular n-gon, central angle θ is 2π/N. The above relations reveal that for large values of N, the inradius R_{i} asymptotically tends to the cirmumradius R_{c} (because θ approaches zero, and the cosine of the third equation approaches unity). It can be visualized that for large N's, the polygon looks very circular in shape, and therefore its circumcircle and incircle match more perfectly with it and as a result with each other.

The area of any regular n-gon can be found, considering that the whole shape can be decomposed to N number of identical isosceles triangles (sides α,R_{c} ,R_{c} ). The area of each triangle is \(\frac{1}{2}a R_i\). Therefore for the polygon:

\[ A = \frac{N}{2} a R_i = \frac{N a^2}{4\tan{\frac{\theta}{2}}} \]

The perimeter of any n-gon is simply the sum of the lengths of all sides:

\[ P = Na \]

### List of convex regular n-gon properties

In the following table, the properties of some key n-gons are listed.

N | φ(°) | θ(°) | Rc/α | Ri/α | Ri/Rc | A/α² |
---|---|---|---|---|---|---|

3 | ||||||

4 | ||||||

5 | ||||||

6 | ||||||

7 | ||||||

8 | ||||||

9 | ||||||

10 | ||||||

11 | ||||||

12 | ||||||

13 | ||||||

14 | ||||||

15 | ||||||

16 | ||||||

20 | ||||||

30 | ||||||

40 | ||||||

50 | ||||||

100 | ||||||

200 | ||||||

1000 |