Maximum area of rectangle inscribed in an equilateral triangle

Given an integer A, which denotes the side of an equilateral triangle, the task is to find the maximum area of the rectangle that can be inscribed in the triangle.

Examples:

Input: A = 10
Output: 21.65
Explanation:
Maximum area of rectangle inscribed in an equilateral triangle of side 10 is 21.65.

Input: A = 12
Output: 31.176
Explanation:
Maximum area of rectangle inscribed in an equilateral triangle of side 12 is 31.176.

Approach: The idea is to use the fact that interior angles of an equilateral triangle is 60o. Then, Draw the perpendicular from one of the side of the triangle and compute the sides of the rectangle with the help of below formulae



The length of Rectangle = (Side of Equilateral Triangle)/2

The breadth of Rectangle = sqrt(3) * (Side of Equilateral Triangle)/4

Then, Maximum area of the rectangle will be \sqrt 3 * \frac {A^2}{8}

Below is the implementation of the above approach:

C++

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// CPP implementation to find the
// maximum area inscribed in an
// equilateral triangle
#include<bits/stdc++.h>
using namespace std;
  
// Function to find the maximum area
// of the rectangle inscribed in an
// equilateral triangle of side S
double solve(int s)
{
    // Maximum area of the rectangle
    // inscribed in an equilateral
    // triangle of side S
    double area = (1.732 * pow(s, 2))/8;
    return area;
      
}
      
// Driver Code
int main()
{
    int n = 14;
    cout << solve(n);
}
      
// This code is contributed by Surendra_Gangwar

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Java

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// Java implementation to find the
// maximum area inscribed in an
// equilateral triangle
  
class GFG
{
    // Function to find the maximum area
    // of the rectangle inscribed in an
    // equilateral triangle of side S
    static double solve(int s)
    {
        // Maximum area of the rectangle
        // inscribed in an equilateral
        // triangle of side S
        double area = (1.732 * Math.pow(s, 2))/8;
        return area;
      
    }
      
    // Driver Code
    public static void  main(String[] args)
    {
        int n = 14;
        System.out.println(solve(n));
    }
}
      
// This article is contributed by Apurva raj

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Python3

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# Python3 implementation to find the
# maximum area inscribed in an
# equilateral triangle
  
# Function to find the maximum area
# of the rectangle inscribed in an
# equilateral triangle of side S
def solve(s):
      
    # Maximum area of the rectangle
    # inscribed in an equilateral
    # triangle of side S
    area = (1.732 * s**2)/8
    return area
      
      
# Driver Code
if __name__=='__main__':
    n = 14
    print(solve(n))

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C#

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// C# implementation to find the
// maximum area inscribed in an
// equilateral triangle
using System;
  
class GFG
{
    // Function to find the maximum area
    // of the rectangle inscribed in an
    // equilateral triangle of side S
    static double solve(int s)
    {
        // Maximum area of the rectangle
        // inscribed in an equilateral
        // triangle of side S
        double area = (1.732 * Math.Pow(s, 2))/8;
        return area;
       
    }
       
    // Driver Code
    public static void  Main(String[] args)
    {
        int n = 14;
        Console.WriteLine(solve(n));
    }
}
  
// This code is contributed by Rajput-Ji

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Output:

42.434

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