# Minimum side of square embedded in Regular polygon with N sides

• Difficulty Level : Hard
• Last Updated : 11 May, 2022

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:
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 ``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;``}`

## 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`

## Python3

 `# Python 3 program to find the minimum``# side of the square in which``# a regular polygon with even sides``# can completely embed``import` `math` `# PI value in Python 3 using``# acos function``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``def` `nGon(N):` `    ``# Projection angle variation``    ``# from axes``    ``proAngleVar ``=` `0` `    ``# 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``    ``negX ``=` `1.0e``+``99``    ``posX ``=` `-``1.0e``+``99``    ``negY ``=` `1.0e``+``99``    ``posY ``=` `-``1.0e``+``99` `    ``for` `j ``in` `range``(N):` `        ``# Projection from all N points``        ``# on X-axis``        ``px ``=` `math.cos(``2` `*` `pi ``*` `j ``/` `N``                      ``+` `proAngleVar)` `        ``# Projection from all N points``        ``# on Y-axis``        ``py ``=` `math.sin(``2` `*` `pi ``*` `j ``/` `N``                      ``+` `proAngleVar)` `        ``negX ``=` `min``(negX, px)``        ``posX ``=` `max``(posX, px)``        ``negY ``=` `min``(negY, py)``        ``posY ``=` `max``(posY, py)` `    ``# Maximum side``    ``opt2 ``=` `max``(posX ``-` `negX,``               ``posY ``-` `negY)` `    ``# Return the portion of side``    ``# forming the square``    ``return` `(opt2 ``/` `math.sin(pi ``/` `N) ``/` `2``)` `# Driver code``if` `__name__ ``=``=` `"__main__"``:` `    ``N ``=` `10``    ``print``(``'%.5f'``%``nGon(N))` `    ``# This code is contributed by ukasp.`

## 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`

## Javascript

 ``

Output:

`3.19623`

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

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