# Area of the Largest Triangle inscribed in a Hexagon

Given here is a regular hexagon, of side length **a**, the task is to find the area of the biggest triangle that can be inscribed within it.**Examples:**

Input:a = 6Output:area = 46.7654Input:a = 8Output:area = 83.1384

**Approach**:

It is very clear that the biggest triangle that can be inscribed within the hexagon is an equilateral triangle.

In triangleACD,

following Pythagoras theorem,(a/2)^2 + (b/2)^2 = a^2b^2/4 = 3a^2/4So, b = a√3

Therefore, area of the triangle,A = √3(a√3)^2/4= 3√3a^2/4

Below is the implementation of the above approach:

## C++

`// C++ Program to find the biggest triangle` `// which can be inscribed within the hexagon` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to find the area` `// of the triangle` `float` `trianglearea(` `float` `a)` `{` ` ` `// side cannot be negative` ` ` `if` `(a < 0)` ` ` `return` `-1;` ` ` `// area of the triangle` ` ` `float` `area = (3 * ` `sqrt` `(3) * ` `pow` `(a, 2)) / 4;` ` ` `return` `area;` `}` `// Driver code` `int` `main()` `{` ` ` `float` `a = 6;` ` ` `cout << trianglearea(a) << endl;` ` ` `return` `0;` `}` |

## Java

`// Java Program to find the biggest triangle` `// which can be inscribed within the hexagon` `import` `java.io.*;` `class` `GFG {` ` ` `// Function to find the area` `// of the triangle` `static` `double` `trianglearea(` `double` `a)` `{` ` ` `// side cannot be negative` ` ` `if` `(a < ` `0` `)` ` ` `return` `-` `1` `;` ` ` `// area of the triangle` ` ` `double` `area = (` `3` `* Math.sqrt(` `3` `) * Math.pow(a, ` `2` `)) / ` `4` `;` ` ` `return` `area;` `}` ` ` `public` `static` `void` `main (String[] args) {` ` ` `double` `a = ` `6` `;` ` ` `System.out.println (trianglearea(a));` ` ` `}` `//This Code is contributed by Sachin..` ` ` `}` |

## Python3

`# Python3 Program to find the biggest triangle` `# which can be inscribed within the hexagon` `import` `math` `# Function to find the area` `# of the triangle` `def` `trianglearea(a):` ` ` `# side cannot be negative` ` ` `if` `(a < ` `0` `):` ` ` `return` `-` `1` `;` ` ` `# area of the triangle` ` ` `area ` `=` `(` `3` `*` `math.sqrt(` `3` `) ` `*` `math.` `pow` `(a, ` `2` `)) ` `/` `4` `;` ` ` `return` `area;` `# Driver code` `a ` `=` `6` `;` `print` `(trianglearea(a))` `# This code is contributed` `# by Akanksha Rai` |

## C#

`// C# Program to find the biggest triangle` `// which can be inscribed within the hexagon` `using` `System;` `class` `GFG {` ` ` `// Function to find the area` `// of the triangle` `static` `double` `trianglearea(` `double` `a)` `{` ` ` `// side cannot be negative` ` ` `if` `(a < 0)` ` ` `return` `-1;` ` ` `// area of the triangle` ` ` `double` `area = (3 * Math.Sqrt(3) * Math.Pow(a, 2)) / 4;` ` ` `return` `Math.Round(area,4);` `}` ` ` `public` `static` `void` `Main () {` ` ` `double` `a = 6;` ` ` `Console.WriteLine(trianglearea(a));` ` ` `}` ` ` `// This code is contributed by Ryuga` `}` |

## PHP

`<?php` `// PHP Program to find the biggest triangle` `// which can be inscribed within the hexagon` `// Function to find the area` `// of the triangle` `function` `trianglearea(` `$a` `)` `{` ` ` `// side cannot be negative` ` ` `if` `(` `$a` `< 0)` ` ` `return` `-1;` ` ` `// area of the triangle` ` ` `$area` `= (3 * sqrt(3) *` ` ` `pow(` `$a` `, 2)) / 4;` ` ` `return` `$area` `;` `}` `// Driver code` `$a` `= 6;` `echo` `trianglearea(` `$a` `);` `// This code is contributed` `// by inder_verma` `?>` |

## Javascript

`<script>` `// javascript Program to find the biggest triangle` `// which can be inscribed within the hexagon` ` ` `// Function to find the area` `// of the triangle` `function` `trianglearea(a)` `{` ` ` `// side cannot be negative` ` ` `if` `(a < 0)` ` ` `return` `-1;` ` ` `// area of the triangle` ` ` `var` `area = (3 * Math.sqrt(3) * Math.pow(a, 2)) / 4;` ` ` `return` `area.toFixed(4);` `}` `var` `a = 6;` `document.write(trianglearea(a));` `// This code contributed by Princi Singh` `</script>` |

**Output:**

46.7654