Area of a square inscribed in a circle which is inscribed in an equilateral triangle

Given here is an equilateral triangle with side length a, which inscribes a circle, which in turn inscribes a square. The task is to find the area of this square.

Examples:

Input: a = 6
Output: 1

Input: a = 10
Output: 0.527046


Approach:

let r be the radius of circle,
hence it is the inradius of equilateral triangle, so r = a /(2 * √3)
diagonal of square, d = diameter of circle = 2 * r = a/ √3
So, area of square, A = 0.5 * d * d
hence A = (1/2) * (a^2) / (3) = (a^2/6)

Below is the implementation of the above approach:

C++

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// C++ Program to find the area of the square
// inscribed within the circle which in turn
// is inscribed in an equilateral triangle
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the area of the square
float area(float a)
{
  
    // a cannot be negative
    if (a < 0)
        return -1;
  
    // area of the square
    float area = sqrt(a) / 6;
  
    return area;
}
  
// Driver code
int main()
{
    float a = 10;
    cout << area(a) << endl;
    return 0;
}

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Java

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// Java Program to find the area of the square
// inscribed within the circle which in turn
// is inscribed in an equilateral triangle
  
import java.io.*;
  
class GFG {
     
  
// Function to find the area of the square
static float area(float a)
{
  
    // a cannot be negative
    if (a < 0)
        return -1;
  
    // area of the square
    float area = (float)Math.sqrt(a) / 6;
  
    return area;
}
  
// Driver code
    public static void main (String[] args) {
        float a = 10;
    System.out.println( area(a));
// This code is contributed 
// by  inder_verma..
    }
}

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Python 3

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# Python3 Program to find the area 
# of the square inscribed within  
# the circle which in turn is 
# inscribed in an equilateral triangle 
  
# import everything from math lib.
from math import *
  
# Function to find the area 
# of the square 
def area(a):
  
    # a cannot be negative 
    if a < 0 :
        return -1
  
    # area of the square 
    area = sqrt(a) / 6
  
    return area
  
# Driver code     
if __name__ == "__main__" :
  
    a = 10
    print(round(area(a), 6))
  
# This code is contributed by ANKITRAI1

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C#

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// C# Program to find the area 
// of the square inscribed within 
// the circle which in turn is 
// inscribed in an equilateral triangle
using System;
  
class GFG 
{
      
// Function to find the area 
// of the square
static float area(float a)
{
  
    // a cannot be negative
    if (a < 0)
        return -1;
  
    // area of the square
    float area = (float)Math.Sqrt(a) / 6;
  
    return area;
}
  
// Driver code
public static void Main ()
{
    float a = 10;
    Console.WriteLine(area(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 square inscribed within 
// the circle which in turn is
// inscribed in an equilateral triangle
  
// Function to find the
// area of the square
function area($a)
{
  
    // a cannot be negative
    if ($a < 0)
        return -1;
  
    // area of the square
    $area = sqrt($a) / 6;
  
    return $area;
}
  
// Driver code
$a = 10;
echo area($a);
  
// This code is contributed 
// by inder_verma
?>

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Output:

0.527046


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Improved By : inderDuMCA, AnkitRai01