# Area of the Largest square 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 rectangle that can be inscribed in it.
Examples:

Input: a = 4, b = 2
Output: 1.25

Input: a = 5, b= 3
Output: 0.604444

Approach: If a square is inscribed in an ellipse, the distance from the centre of the square to any of its corners will be equal to the distance between the origin and the point on the upper right corner in the diagram below, where x=y

the equation of the ellipse is x^2/a^2 + y^2/b^2 = 1
If, x = y
then, x^2/a^2 + x^2/b^2 = 1
therefore, x = ?(a^2 + b^2)/ab
so, y = ?(a^2 + b^2)/ab
So Area, A = 4(a^2 + b^2)/a^2b^2

Below is the implementation of above approach:

## C++

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

## Java

 // Java Program to find the biggest square // which can be inscribed within the ellipse import java.io.*;   class GFG {     // Function to find the area // of the square static float squarearea(float a, float b) {       // a and b cannot be negative     if (a < 0 || b < 0)         return -1;       // area of the square     float area = 4 *(float) ((Math.pow(a, 2) + Math.pow(b, 2))                     / (Math.pow(a, 2) * Math.pow(b, 2)));       return area; }   // Driver code       public static void main (String[] args) {         float a = 4, b = 2;     System.out.println( squarearea(a, b));       } } // This code is contributed by inder_verma.

## Python 3

 # Python3 Program to find the biggest square # which can be inscribed within the ellipse     # Function to find the area # of the square def squarearea( a, b):         # a and b cannot be negative     if (a < 0 or b < 0):         return -1             # area of the square     area = 4 * (((pow(a, 2) + pow(b, 2)) /                (pow(a, 2) * pow(b, 2))))       return area     # Driver code if __name__=='__main__':     a = 4     b = 2     print(squarearea(a, b))   # This code is contributed by ash264

## C#

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



## Javascript



Output:

1.25

Time complexity: O(1)

Auxiliary Space: O(1)

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