# Area of largest Circle inscribe in N-sided Regular polygon

Given a regular polygon of N sides with side length a. The task is to find the area of the Circle which inscribed in the polygon.

Note : This problem is mixed version of This and This
Examples:

```Input: N = 6, a = 4
Output: 37.6801
Explanataion:

In this, the polygon have 6 faces
and as we see in fig.1 we clearly see
that the angle  x  is 30 degree
so the radius of circle will be ( a / (2  * tan(30)))
Therefore, r = a√3/2

Input: N = 8, a = 8
Output: 292.81
Explanataion:

In this, the polygon have 8 faces
and as we see in fig.2 we clearly see
that the angle  x  is 22.5 degree
so the radius of circle will be ( a / (2  * tan(22.5)))
Therefore, r = a/0.828
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: In the figure above, we see the polygon can be divided into N equal triangles. Looking into one of the triangles, we see that the whole angle at the center can be divided into = 360/N

So, angle x = 180/n
Now, tan(x) = (a / 2) * r

So, r = a / ( 2 * tan(x))

So, Area of the Inscribed Circle is,

``` A = Πr² = Π * (a / (2 * tan(x))) * (a / (2*tan(x)))
```

Below is the implementation of the above approach:

## C++

 `// C++ Program to find the area of a circle in ` `// inscribed in polygon ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to find the area ` `// of a circle ` `float` `InscribedCircleArea(``float` `n, ``float` `a) ` `{ ` `    ``// Side and side length cannot be negative ` `    ``if` `(a < 0 && n < 0) ` `        ``return` `-1; ` ` `  `    ``// degree converted to radians ` `    ``float` `r = a / (2 * ``tan``((180 / n) * 3.14159 / 180)); ` ` `  `    ``// area of circle ` `    ``float` `Area = (3.14) * (r) * (r); ` ` `  `    ``return` `Area; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` ` `  `    ``// no.  of sides ` `    ``float` `n = 6; ` ` `  `    ``// side length ` `    ``float` `a = 4; ` ` `  `    ``cout << InscribedCircleArea(n, a) << endl; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java Program to find the area of a circle ` `// inscribed in a polygon ` `import` `java.io.*; ` ` `  `class` `GFG { ` ` `  `    ``// Function to find the area ` `    ``// of a regular polygon ` `    ``static` `float` `InscribedCircleArea(``float` `n, ``float` `a) ` `    ``{ ` `        ``// Side and side length cannot be negative ` `        ``if` `(a < ``0` `&& n < ``0``) ` `            ``return` `-``1``; ` ` `  `        ``// degree converted to radians ` `        ``float` `r = a / (``float``)(``2` `* Math.tan((``180` `/ n) * ``3.14159` `/ ``180``)); ` ` `  `        ``// area of circle ` `        ``float` `Area = (``float``)(``3.14``) * (r) * (r); ` ` `  `        ``return` `Area; ` `    ``} ` ` `  `    ``// Driver code ` ` `  `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` ` `  `        ``// no.  of sides ` `        ``float` `n = ``6``; ` ` `  `        ``// side length ` `        ``float` `a = ``4``; ` ` `  `        ``System.out.println(InscribedCircleArea(n, a)); ` `    ``} ` `} `

## Python3

 `# Python 3 Program to find the area  ` `# of a circle inscribed ` `# in a polygon ` `from` `math ``import` `tan ` ` `  `# Function to find the area of a  ` `# circle ` `def` `InscribedCircleArea(n, a): ` `    ``# Side and side length cannot  ` `    ``# be negative ` `    ``if` `(a < ``0` `and` `n < ``0``): ` `        ``return` `-``1` ` `  `    ``# degree converted to radians ` `    ``r ``=` `a``/``(``2` `*` `tan((``180` `/` `n) ``*` `3.14159` `/` `180``)); ` ` `  `    ``# area of circle ` `    ``Area ``=` `3.14` `*` `r ``*` `r ` ` `  `    ``return` `Area ` ` `  `# Driver code ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``a ``=` `4` `    ``n ``=` `6` ` `  `    ``print``(``'{0:.6}'``.``format``(InscribedCircleArea(n, a))) ` ` `  `# This code is contributed by ` `# Chandan Agrawal `

## C#

 `// C# Program to find the area of a circle  ` `// inscribed in a polygon  ` `using` `System; ` ` `  `class` `GFG  ` `{  ` ` `  `// Function to find the area  ` `// of a regular polygon  ` `static` `float` `InscribedCircleArea(``float` `n, ``float` `a)  ` `{  ` `    ``// Side and side length cannot be negative  ` `    ``if` `(a < 0 && n < 0)  ` `        ``return` `-1;  ` ` `  `    ``// degree converted to radians  ` `    ``float` `r = a / (``float``)(2 * Math.Tan((180 / n) *  ` `                                 ``3.14159 / 180));  ` ` `  `    ``// area of circle  ` `    ``float` `Area = (``float``)(3.14) * (r) * (r);  ` ` `  `    ``return` `Area;  ` `}  ` ` `  `// Driver code  ` `public` `static` `void` `Main()  ` `{  ` ` `  `    ``// no. of sides  ` `    ``float` `n = 6;  ` ` `  `    ``// side length  ` `    ``float` `a = 4;  ` ` `  `    ``Console.WriteLine(InscribedCircleArea(n, a));  ` `} ` `}  ` ` `  `// This code is contributed by Ryuga `

## PHP

 ` `

Output:

```37.6801
```

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