# Largest Square that can be inscribed within a hexagon

Given a regular hexagonof side length **a**, the task is to find the area of the largest square that can be inscribed within it.

**Examples:**

Input:a = 6

Output:57.8817

Input:a = 8

Output:102.901

**Approach:** The square we will derive will have the same centre and axes of the hexagon. This is because the square will become smaller if we will rotate it.

The sides of the hexagonal are equal i.e.

a = b + c.

Now, letdbe 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 ofbandain the left hand side equation of square, we get,

d / a = 3 – √3i.e.d / a = 1.268

Therefore,d = 1.268 * a

Below is the implementation of the above approach:

## C++

`// C++ program to find the area of the largest square ` `// that can be inscribed within the hexagon ` `#include <bits/stdc++.h> ` `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; ` `} ` |

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## 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)); ` ` ` `} ` `} ` `//contributed by Arnab Kundu ` |

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**Output:**

57.8817

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