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

Namachila November 19, 2025

0 0 3 minutes read

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++

// 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;
}

Java

// 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..
    }
}

Python 3

# 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

C#

// 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

PHP

<?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
?>

Javascript

<script>
// javascript 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
    var area = Math.sqrt(a) / 6;
 
    return area;
}
 
// Driver code
var a = 10;
document.write( area(a).toFixed(6));
 
 
// This code contributed by shikhasingrajput 
 
</script>

Output:

0.527046

Time complexity : O(log(a)) for given side a, as complexity of inbuilt sqrt function
Auxiliary Space : O(1)

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