可以刻在六边形内的最大正方形

给定边长为a的正六边形，任务是找到可以在其中内接的最大正方形的面积。
例子：

Input: a = 6 
Output: 57.8817
Input: a = 8 
Output: 102.901 
 

方法：我们将得出的正方形将具有与六边形相同的中心和轴。这是因为如果旋转正方形，正方形将变小。

The sides of the hexagonal are equal i.e. a = b + c. 
Now, let d be the length of the side of the inscribed square, 
Then the top side of the square, d = 2 * c * sin(60). 
And, the left side of the square, d = a + 2 * b * sin(30). 
Substituting for c, d = 2 * (a – b) * sin(60). 
Now taking d and re-arranging, we get, b / a = (2 * sin(60) – 1) / (2 * (sin(30) + sin(60))) 
So, b / a = 2 – √3 
Now, substituting the relation of b and a in the left hand side equation of square, we get, 
d / a = 3 – √3 i.e. d / a = 1.268 
Therefore, d = 1.268 * a 
 

下面是上述方法的实现：

C++

// C++ program to find the area of the largest square
// that can be inscribed within the hexagon
#include 
using namespace std;
 
// Function to find the area
// of the square
float squareArea(float a)
{
 
    // Side cannot be negative
    if (a < 0)
        return -1;
 
    // Area of the square
    float area = pow(1.268, 2) * pow(a, 2);
    return area;
}
 
// Driver code
int main()
{
    float a = 6;
    cout << squareArea(a) << endl;
    return 0;
}

Java

// Java program to find the area of the largest square
// that can be inscribed within the hexagon
class Solution {
    // Function to find the area
    // of the square
    static float squareArea(float a)
    {
 
        // Side cannot be negative
        if (a < 0)
            return -1;
 
        // Area of the square
        float area = (float)(Math.pow(1.268, 2) * Math.pow(a, 2));
        return area;
    }
 
    // Driver code
    public static void main(String args[])
    {
        float a = 6;
        System.out.println(squareArea(a));
    }
}
 
// This code is contributed by Arnab Kundu

Python3

# Python program to find the area of the largest square
# that can be inscribed within the hexagon
 
# Function to find the area
# of the square
def squareArea(a):
 
    # Side cannot be negative
    if (a < 0):
        return -1;
 
    # Area of the square
    area = (1.268 ** 2) * (a ** 2);
    return area;
 
# Driver code
a = 6;
print(squareArea(a));
 
# This code contributed by PrinciRaj1992

C#

// C# program to find the area of the largest square
// that can be inscribed within the hexagon
using System;
class Solution {
    // Function to find the area
    // of the square
    static float squareArea(float a)
    {
 
        // Side cannot be negative
        if (a < 0)
            return -1;
 
        // Area of the square
        float area = (float)(Math.Pow(1.268, 2) * Math.Pow(a, 2));
        return area;
    }
 
    // Driver code
    public static void Main()
    {
        float a = 6;
        Console.WriteLine(squareArea(a));
    }
}
 
// This code is contributed by  anuj_67..

Javascript

// javascript program to find the area of the largest square
// that can be inscribed within the hexagon
 
// Function to find the area
// of the square
function squareArea(a)
{
 
  // Side cannot be negative
  if (a < 0)
  return -1;
 
  // Area of the square
  var area = (Math.pow(1.268, 2) * Math.pow(a, 2));
  return area;
}
 
// Driver code
 
var a = 6;
document.write(squareArea(a).toFixed(5));
 
// This code is contributed by Princi Singh

输出：

57.8817