# 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 = 4Output: 37.6801Explanataion:In this, the polygon have 6 faces and as we see in fig.1 we clearly see that the anglexis 30 degree so the radius of circle will be ( a / (2 * tan(30))) Therefore,r = a√3/2Input:N = 8, a = 8Output:292.81Explanataion:In this, the polygon have 8 faces and as we see in fig.2 we clearly see that the anglexis 22.5 degree so the radius of circle will be ( a / (2 * tan(22.5))) Therefore,r = a/0.828

**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 <bits/stdc++.h> ` `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; ` `} ` |

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## 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)); ` ` ` `} ` `} ` |

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## 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 ` |

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## 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 ` |

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## PHP

`<?php ` `// PHP Program to find the area ` `// of a circle inscribed ` `// in a polygon ` ` ` `// Function to find the area of a ` `// circle ` `function` `InscribedCircleArea(` `$n` `, ` `$a` `) ` `{ ` ` ` `// Side and side length cannot ` ` ` `// be negative ` ` ` `if` `(` `$a` `< 0 && ` `$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` `$Area` `; ` `} ` ` ` `// Driver code ` `$a` `= 4; ` `$n` `= 6; ` `echo` `(InscribedCircleArea(` `$n` `, ` `$a` `)); ` ` ` `// This code contributed by PrinciRaj1992 ` `?> ` |

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**Output:**

37.6801

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