Área de un círculo inscrito en un hexágono regular

Dado un Hexágono regular de lado a , la tarea es encontrar el área del círculo inscrito en él, dado que el círculo es tangente a cada uno de los seis lados.
Ejemplos: 
 

Input: a = 4
Output: 37.68

Input: a = 10
Output: 235.5

Enfoque : 
De la figura, está claro que podemos dividir el hexágono regular en 6 triángulos equiláteros idénticos. 
Tomamos un triángulo OAB , con O como el centro del hexágono o círculo, y AB como un lado del hexágono. 
Sea M el punto medio de AB , OM sería la bisectriz perpendicular de AB , ángulo AOM = 30 grados
Luego, en el triángulo rectángulo OAM,
 

tanx = tan30 = 1/√3 
Entonces, a/2r = 1/√3 
Por lo tanto, r = a√3/2 
Área del círculo, A =Πr²=Π3a^2/4

A continuación se muestra la implementación del enfoque : 
 

// C++ Program to find the area of the circle
// which can be inscribed within the hexagon
#include <bits/stdc++.h>
using namespace std;
 
// Function to find the area
// of the inscribed circle
float circlearea(float a)
{
 
    // the side cannot be negative
    if (a < 0)
        return -1;
 
    // area of the circle
    float A = (3.14 * 3 * pow(a, 2)) / 4;
 
    return A;
}
 
// Driver code
int main()
{
    float a = 4;
    cout << circlearea(a) << endl;
 
    return 0;
}

//Java program to find the
//area of the circle
//which can be inscribed within the hexagon
 
import java.util.*;
 
class solution
{
static double circlearea(double a)
{
 
// the side cannot be negative
    if (a < 0)
    return -1;
 
// area of the circle
    double A = (3.14 * 3 * Math.pow(a,2) ) / 4;
 
    return A;
}
public static void main(String arr[])
{
    double a = 4;
    System.out.println(circlearea(a));
}
}

# Python 3 program to find the
# area of the circle
# which can be inscribed within the hexagon
 
# Function to find the area
# of the inscribed circle
def circlearea(a) :
 
    # the side cannot be negative
    if a < 0 :
        return -1
 
    #  area of the circle
    A = (3.14 * 3 * pow(a,2)) / 4
 
    return A
 
 
# Driver code    
if __name__ == "__main__" :
 
    a = 4
    print(circlearea(a))
 
 
# This code is contributed by ANKITRAI1

// C# program to find 
// the area of the circle
// which can be inscribed
// within the hexagon
using System;
 
class GFG
{
static double circlearea(double a)
{
 
    // the side cannot be negative
    if (a < 0)
    return -1;
 
    // area of the circle
    double A = (3.14 * 3 *
                Math.Pow(a, 2)) / 4;
 
    return A;
}
 
// Driver Code
public static void Main()
{
    double a = 4;
    Console.WriteLine(circlearea(a));
}
}
 
// This code is contributed
// by inder_verma

<?php
// PHP Program to find the area of
// the circle which can be inscribed
// within the hexagon
 
// Function to find the area
// of the inscribed circle
function circlearea($a)
{
 
    // the side cannot be negative
    if ($a < 0)
        return -1;
 
    // area of the circle
    $A = (3.14 * 3 * pow($a, 2)) / 4;
 
    return $A;
}
 
// Driver code
$a = 4;
echo circlearea($a) . "\n";
 
// This code is contributed
// by Akanksha Rai(Abby_akku)

<script>
 
// javascript program to find the
//area of the circle
//which can be inscribed within the hexagon
 
function circlearea(a) {
 
    // the side cannot be negative
    if (a < 0)
        return -1;
 
    // area of the circle
    var A = (3.14 * 3 * Math.pow(a, 2)) / 4;
 
    return A;
}
 
var a = 4;
document.write(circlearea(a));
 
// This code is contributed by 29AjayKumar
 
</script>

37.68

Tiempo Complejidad: O(1)
Espacio Auxiliar: O(1)