Largest triangle that can be inscribed in an ellipse

Given an ellipse, with major axis length 2a & 2b, the task is to find the area of the largest triangle that can be inscribed in it.

Examples:

Input: a = 4, b = 2
Output: 10.3923

Input: a = 5, b = 3
Output: 10.8253



Approach: So we know the ellipse is just the scaled shadow of a circle.Let’s find the scaling factor.

x^2/a^2 + y^2/b^2 = 1 is an ellipse. Rewrite this as:
(y*(a/b))^2+x^2 = a^2

This is just a vertically scaled down circle of radius a (think light falls from the top at an angle), and the vertical factor is a/b. The biggest triangle in the ellipse is then a scaled up version of the biggest triangle in the circle. Using a little geometry and taking symmetry into account, we can understand that the biggest such triangle is the equilateral one. It’s sides will be √3a and the area will be (3√3)a^2/4

Translating this to ellipse terms – we scale the horizontal dimension up by a factor a/b, and the area of the biggest triangle in the ellipse is,

A = (3√3)a^2/4b

Below is the implementation of above approach:

C++

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// C++ Program to find the biggest triangle
// which can be inscribed within the ellipse
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the area
// of the triangle
float trianglearea(float a, float b)
{
  
    // a and b cannot be negative
    if (a < 0 || b < 0)
        return -1;
  
    // area of the triangle
    float area = (3 * sqrt(3) * pow(a, 2)) / (4 * b);
  
    return area;
}
  
// Driver code
int main()
{
    float a = 4, b = 2;
    cout << trianglearea(a, b) << endl;
  
    return 0;
}

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Java

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//Java Program to find the biggest triangle
//which can be inscribed within the ellipse
  
public class GFG {
  
    //Function to find the area
    //of the triangle
    static float trianglearea(float a, float b)
    {
  
     // a and b cannot be negative
     if (a < 0 || b < 0)
         return -1;
  
     // area of the triangle
     float area = (float)(3 * Math.sqrt(3) * Math.pow(a, 2)) / (4 * b);
  
     return area;
    }
  
    //Driver code
    public static void main(String[] args) {
      
        float a = 4, b = 2;
         System.out.println(trianglearea(a, b));
    }
}

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Python3

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# Python 3 Program to find the biggest triangle 
# which can be inscribed within the ellipse
  
from math import *
  
# Function to find the area 
# of the triangle
def trianglearea(a, b) :
  
    # a and b cannot be negative 
    if a < 0 or b < 0 :
        return -1
  
    # area of the triangle 
    area = (3 * sqrt(3) * pow(a, 2)) / (4 * b)
  
    return area
  
  
# Driver Code
if __name__ == "__main__" :
  
    a, b = 4, 2
    print(round(trianglearea(a, b),4))
  
  
# This code is contributed by ANKITRAI1

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C#

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// C# Program to find the biggest 
// triangle which can be inscribed
// within the ellipse
using System;
  
class GFG
{
  
// Function to find the area
// of the triangle
static float trianglearea(float a, float b)
{
  
// a and b cannot be negative
if (a < 0 || b < 0)
    return -1;
  
// area of the triangle
float area = (float)(3 * Math.Sqrt(3) *
                         Math.Pow(a, 2)) / (4 * b);
  
return area;
}
  
// Driver code
public static void Main()
{
    float a = 4, b = 2;
    Console.WriteLine(trianglearea(a, b));
}
}
  
// This code is contributed 
// by Akanksha Rai(Abby_akku)

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PHP

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<?php
// PHP Program to find the biggest
// triangle which can be inscribed 
// within the ellipse
  
// Function to find the area
// of the triangle
function trianglearea($a, $b)
{
  
    // a and b cannot be negative
    if ($a < 0 || $b < 0)
        return -1;
  
    // area of the triangle
    $area = (3 * sqrt(3) * 
            pow($a, 2)) / (4 * $b);
  
    return $area;
}
  
// Driver code
$a = 4;
$b = 2;
echo trianglearea($a, $b); 
  
// This code is contributed 
// by Shivi_Aggarwal 
?>

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Output:

10.3923


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