Area of a circle inscribed in a regular hexagon

Given a regular Hexagon with side length a, the task is to find the area of the circle inscribed in it, given that, the circle is tangent to each of the six sides.

Examples:

Input: a = 4
Output: 37.68

Input: a = 10
Output: 235.5

Approach:
From the figure, it is clear that, we can divide the regular hexagon into 6 identical equilateral triangles.
We take one triangle OAB, with O as the centre of the hexagon or circle, & AB as one side of the hexagon.
Let M be mid-point of AB, OM would be the perpendicular bisector of AB, angle AOM = 30 deg

Then in right angled triangle OAM,

tanx = tan30 = 1/√3
So, a/2r = 1/√3
Therefore, r = a√3/2
Area of circle, A =Πr²=Π3a^2/4

Below is the implementation of the approach:

C++

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// 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;
}

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Java

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//Java program to 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));
}
}

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Python 3

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# 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

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C#

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// C# program to 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

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PHP

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<?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)

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Output:

37.68


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