Regular and Irregular Polygons

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Irregular convex polygon

Sum of interior angles of an n-sided convex polygon:

$$\sum _{i=1}^n{\alpha }_i={\alpha }_1+{\alpha }_2+{\alpha }_3+ \cdots +{\alpha }_n=\left(n-2\right)·180°$$

Sum of exterior angles of an n-sided convex polygon:

$$\sum _{i=1}^n{\alpha }_i^{\text{'}}={\alpha }_1^{\text{'}}+{\alpha }_2^{\text{'}}+{\alpha }_3^{\text{'}}+ \cdots +{\alpha }_n^{\text{'}}=360°$$

Number of diagonals of an n-sided convex polygon:

$$N=\frac{n·\left(n-3\right)}{2}$$

The size of an angle of an n-sided regular polygon:

$$\alpha =\frac{\left(n-2\right)·180°}{\mathrm{n}}$$

The size of the central angle of an n-sided regular polygon:

$$\beta =\frac{360°}{n}$$

Perimeter of a regular polygon with n sides:

$$K=n·a$$

The area of ​​a regular polygon with n sides, based on side and the circumference of the circle inscribed in the polygon:

$$T=\frac{1}{2}·n·a·r$$

Area of ​​a regular polygon with n sides, based on the circumference of the circle inscribed in the polygon and the central angle:

$$T=\frac{n·R^2·\sin \beta }{2}$$

Area of ​​an n-sided regular polygon, based on side and central angle:

$$T=\frac{n·a^2·ctg \frac{\beta }{2}}{4}$$