Biggest Square that can be inscribed within an Equilateral triangle
Given here is an equilateral triangle of side length a. The task is to find the side of the biggest square that can be inscribed within it.
Examples:
Input: a = 5 Output: 2.32 Input: a = 7 Output: 3.248
Approach: Let the side of the square be x.
Now, AH is perpendicular to DE.
DE is parallel to BC, So, angle AED = angle ACB = 60
In triangle EFC, => Sin60 = x/ EC => ?3 / 2 = x/EC => EC = 2x/?3 In triangle AHE, => Cos 60 = x/2AE => 1/2 = x/2AE => AE = x
So, side AC of the triangle = 2x/?3 + x. Now,
a = 2x/?3 + x
Therefore, x = a/(1 + 2/?3) = 0.464a
Below is the implementation of the above approach:
C++
// C++ Program to find the biggest square // which can be inscribed within the equilateral triangle #include <bits/stdc++.h> using namespace std; // Function to find the side // of the square float square( float a) { // the side cannot be negative if (a < 0) return -1; // side of the square float x = 0.464 * a; return x; } // Driver code int main() { float a = 5; cout << square(a) << endl; return 0; } |
Java
// Java Program to find the // the biggest square which // can be inscribed within // the equilateral triangle class GFG { // Function to find the side // of the square static double square( double a) { // the side cannot be negative if (a < 0 ) return - 1 ; // side of the square double x = 0.464 * a; return x; } // Driver code public static void main(String []args) { double a = 5 ; System.out.println(square(a)); } } // This code is contributed by ihritik |
Python3
# Python3 Program to find the biggest square # which can be inscribed within the equilateral triangle # Function to find the side # of the square def square( a ): # the side cannot be negative if (a < 0 ): return - 1 # side of the square x = 0.464 * a return x # Driver code a = 5 print (square(a)) # This code is contributed by ihritik |
C#
// C# Program to find the biggest // square which can be inscribed // within the equilateral triangle using System; class GFG { // Function to find the side // of the square static double square( double a) { // the side cannot be negative if (a < 0) return -1; // side of the square double x = 0.464 * a; return x; } // Driver code public static void Main() { double a = 5; Console.WriteLine(square(a)); } } // This code is contributed by ihritik |
PHP
<?php // PHP Program to find the biggest // square which can be inscribed // within the equilateral triangle // Function to find the side // of the square function square( $a ) { // the side cannot be negative if ( $a < 0) return -1; // side of the square $x = 0.464 * $a ; return $x ; } // Driver code $a = 5; echo square( $a ); // This code is contributed by ihritik ?> |
Javascript
<script> // javascript Program to find the // the biggest square which // can be inscribed within // the equilateral triangle // Function to find the side // of the square function square(a) { // the side cannot be negative if (a < 0) return -1; // side of the square var x = 0.464 * a; return x; } // Driver code var a = 5; document.write(square(a).toFixed(2)); // This code contributed by Princi Singh </script> |
Output:
2.32
Time Complexity: O(1)
Auxiliary Space: O(1)