Given an even number **N** which represents the number of sides of a regular polygon with **N** vertices, the task is to find the square of the minimum size such that given **Polygon** can completely embed in the square.

A Polygon is a convex figure and has equal sides and equal angles. All sides have length

1.

**Embedding:** Place **Polygon** in the square in such way that each point which lies inside or on a border of **N** should also lie inside or on a border of the square.

**Examples:**

Input:N = 4

Output:1

Explanation:

Regular polygon with 4 Sides is square with side 1.

Given polygon can easily embed on the square with side 1.

Input:N = 6

Output:1.931851653

Explanation :

Regular polygon with 6 Sides is Hexagon with side 1.

Given polygon can easily embed on the square with side 1.931851653.

**Approach:** The idea is to observe that on a 3-D plane, when a polygon is embedded in a square, it might be rotated. A similar approach has been discussed in Hexagon problem and Octagon problem . Therefore, we take the projection of each side on both the axis using the mathematical functions *sin()* and *cos()*. The overall sum of all the projections is the minimum side of the square required in this problem.

Below is the implementation of the above approach:

## C++

`// C++ program to find the minimum ` `// side of the square in which ` `// a regular polygon with even sides ` `// can completely embed ` ` ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// PI value in C++ using ` `// acos function ` `const` `double` `pi = ` `acos` `(-1.0); ` ` ` `// Function to find the minimum ` `// side of the square in which ` `// a regular polygon with even sides ` `// can completely embed ` `double` `nGon(` `int` `N) ` `{ ` ` ` ` ` `// Projection angle variation ` ` ` `// from axes ` ` ` `double` `proAngleVar; ` ` ` ` ` `// Projection angle variation ` ` ` `// when the number of ` ` ` `// sides are in multiple of 4 ` ` ` `if` `(N % 4 == 0) { ` ` ` `proAngleVar ` ` ` `= pi * (180.0 / N) ` ` ` `/ 180; ` ` ` `} ` ` ` `else` `{ ` ` ` `proAngleVar ` ` ` `= pi * (180.0 / (2 * N)) ` ` ` `/ 180; ` ` ` `} ` ` ` ` ` `// Distance between the end points ` ` ` `double` `negX = 1.0e+99, ` ` ` `posX = -1.0e+99, ` ` ` `negY = 1.0e+99, ` ` ` `posY = -1.0e+99; ` ` ` ` ` `for` `(` `int` `j = 0; j < N; ++j) { ` ` ` ` ` `// Projection from all N points ` ` ` `// on X-axis ` ` ` `double` `px = ` `cos` `(2 * pi * j / N ` ` ` `+ proAngleVar); ` ` ` ` ` `// Projection from all N points ` ` ` `// on Y-axis ` ` ` `double` `py = ` `sin` `(2 * pi * j / N ` ` ` `+ proAngleVar); ` ` ` ` ` `negX = min(negX, px); ` ` ` `posX = max(posX, px); ` ` ` `negY = min(negY, py); ` ` ` `posY = max(posY, py); ` ` ` `} ` ` ` ` ` `// Maximum side ` ` ` `double` `opt2 = max(posX - negX, ` ` ` `posY - negY); ` ` ` ` ` `// Return the portion of side ` ` ` `// forming the square ` ` ` `return` `(` `double` `)opt2 ` ` ` `/ ` `sin` `(pi / N) / 2; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `int` `N = 10; ` ` ` `cout << nGon(N); ` ` ` `return` `0; ` `} ` |

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## Java

`// Java program to find the minimum ` `// side of the square in which ` `// a regular polygon with even sides ` `// can completely embed ` `class` `GFG{ ` ` ` `// PI value in Java using ` `// acos function ` `static` `double` `pi = Math.acos(-` `1.0` `); ` ` ` `// Function to find the minimum ` `// side of the square in which ` `// a regular polygon with even sides ` `// can completely embed ` `static` `double` `nGon(` `int` `N) ` `{ ` ` ` ` ` `// Projection angle variation ` ` ` `// from axes ` ` ` `double` `proAngleVar; ` ` ` ` ` `// Projection angle variation ` ` ` `// when the number of ` ` ` `// sides are in multiple of 4 ` ` ` `if` `(N % ` `4` `== ` `0` `) ` ` ` `{ ` ` ` `proAngleVar = pi * (` `180.0` `/ N) / ` `180` `; ` ` ` `} ` ` ` `else` ` ` `{ ` ` ` `proAngleVar = pi * (` `180.0` `/ (` `2` `* N)) / ` `180` `; ` ` ` `} ` ` ` ` ` `// Distance between the end points ` ` ` `double` `negX = ` `1` `.0e+` `99` `, ` ` ` `posX = -` `1` `.0e+` `99` `, ` ` ` `negY = ` `1` `.0e+` `99` `, ` ` ` `posY = -` `1` `.0e+` `99` `; ` ` ` ` ` `for` `(` `int` `j = ` `0` `; j < N; ++j) ` ` ` `{ ` ` ` ` ` `// Projection from all N points ` ` ` `// on X-axis ` ` ` `double` `px = Math.cos(` `2` `* pi * j / N + ` ` ` `proAngleVar); ` ` ` ` ` `// Projection from all N points ` ` ` `// on Y-axis ` ` ` `double` `py = Math.sin(` `2` `* pi * j / N + ` ` ` `proAngleVar); ` ` ` ` ` `negX = Math.min(negX, px); ` ` ` `posX = Math.max(posX, px); ` ` ` `negY = Math.min(negY, py); ` ` ` `posY = Math.max(posY, py); ` ` ` `} ` ` ` ` ` `// Maximum side ` ` ` `double` `opt2 = Math.max(posX - negX, ` ` ` `posY - negY); ` ` ` ` ` `// Return the portion of side ` ` ` `// forming the square ` ` ` `return` `(` `double` `)opt2 / ` ` ` `Math.sin(pi / N) / ` `2` `; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main(String[] args) ` `{ ` ` ` `int` `N = ` `10` `; ` ` ` `System.out.printf(` `"%.5f"` `,nGon(N)); ` `} ` `} ` ` ` `// This code is contributed by 29AjayKumar ` |

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## C#

`// C# program to find the minimum ` `// side of the square in which ` `// a regular polygon with even sides ` `// can completely embed ` `using` `System; ` `class` `GFG{ ` ` ` `// PI value in Java using ` `// acos function ` `static` `double` `pi = Math.Acos(-1.0); ` ` ` `// Function to find the minimum ` `// side of the square in which ` `// a regular polygon with even sides ` `// can completely embed ` `static` `double` `nGon(` `int` `N) ` `{ ` ` ` ` ` `// Projection angle variation ` ` ` `// from axes ` ` ` `double` `proAngleVar; ` ` ` ` ` `// Projection angle variation ` ` ` `// when the number of ` ` ` `// sides are in multiple of 4 ` ` ` `if` `(N % 4 == 0) ` ` ` `{ ` ` ` `proAngleVar = pi * (180.0 / N) / 180; ` ` ` `} ` ` ` `else` ` ` `{ ` ` ` `proAngleVar = pi * (180.0 / (2 * N)) / 180; ` ` ` `} ` ` ` ` ` `// Distance between the end points ` ` ` `double` `negX = 1.0e+99, ` ` ` `posX = -1.0e+99, ` ` ` `negY = 1.0e+99, ` ` ` `posY = -1.0e+99; ` ` ` ` ` `for` `(` `int` `j = 0; j < N; ++j) ` ` ` `{ ` ` ` ` ` `// Projection from all N points ` ` ` `// on X-axis ` ` ` `double` `px = Math.Cos(2 * pi * j / N + ` ` ` `proAngleVar); ` ` ` ` ` `// Projection from all N points ` ` ` `// on Y-axis ` ` ` `double` `py = Math.Sin(2 * pi * j / N + ` ` ` `proAngleVar); ` ` ` ` ` `negX = Math.Min(negX, px); ` ` ` `posX = Math.Max(posX, px); ` ` ` `negY = Math.Min(negY, py); ` ` ` `posY = Math.Max(posY, py); ` ` ` `} ` ` ` ` ` `// Maximum side ` ` ` `double` `opt2 = Math.Max(posX - negX, ` ` ` `posY - negY); ` ` ` ` ` `// Return the portion of side ` ` ` `// forming the square ` ` ` `return` `(` `double` `)opt2 / ` ` ` `Math.Sin(pi / N) / 2; ` `} ` ` ` `// Driver code ` `public` `static` `void` `Main() ` `{ ` ` ` `int` `N = 10; ` ` ` `Console.Write(` `string` `.Format(` `"{0:F5}"` `, nGon(N))); ` `} ` `} ` ` ` `// This code is contributed by Nidhi_biet ` |

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**Output:**

3.19623

**Time Complexity:** *O(N)*, where N is number of sides of polygon.

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