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 = 6
Output: area = 46.7654

Input: a = 8
Output: area = 83.1384

Approach:

It is very clear that the biggest triangle that can be inscribed within the hexagon is an equilateral triangle. 
In triangle ACD, 
following Pythagoras theorem, 
(a/2)^2 + (b/2)^2 = a^2 
b^2/4 = 3a^2/4 
So, 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

Time complexity: O(1)

Auxiliary Space: O(1)