The area of the circle entered in a polygon can be calculated not only through parameters of the circle, but through various elements of the described figure - the parties, height, diagonals, perimeter.

## Instruction

1. The circle is called entered in a polygon if has the general point with each party of the described figure. The center of the circle entered in a polygon always lies in a point of intersection of bisectors of its internal corners. The area limited to a circle is defined by a formula S=π*r² where r is circle radius, π - number of Pi - the mathematical constant equal 3.14. For the circle entered in a geometrical figure, radius is equal to a piece from the center to a contact point with the party of a figure. Therefore, it is possible to define dependence between the radius of the circle entered in a polygon and elements of this figure and to express the area of a circle through parameters of the described polygon.

2. It is possible to enter the only circle in any triangle with a radius determined by a formula: r=s /p ∆ where r is the radius of an inscribed circle, s ∆ - the area of a triangle, p ∆ - poluperimetr a triangle. Substitute the received value of radius expressed through elements of the triangle described about a circle in a formula of the area of a circle. Then the area S circles, ∆ and poluperimetry p ∆ is calculated by s, inscribed in a triangle with an area, on a formula: S = π * (s /p ∆)².

3. The circle can be entered in a convex quadrangle provided that the sums of the opposite parties are equal in it. The area S circles, entered in a square with the party of a, is equal: S = π*a²/4.

4. The area S inscribed circles is equal in a rhombus: S = π * (d₁d ₂/4a)². In this formula d ₁ and d ₂ — rhombus diagonals, and - the party of a rhombus. For a trapeze the area of S of the circle entered in it is determined by a formula: S = π * (h/2)² where h is trapeze height.

5. The party and the correct hexagon is equal to the radius of the circle entered in it, the area S circles is calculated on a formula: S = π*a². A circle it is possible to enter in a regular polygon with any number of the parties. The general formula for determination of radius of r of the circle entered in a polygon with the party and and number of the parties of n: r=a/2tg (360 °/2n). The area of S of the circle entered in such polygon: S=π * (a/2tg (360 °/2n)²/2.