Largest square that can be inscribed in a semicircle
Given a semicircle with radius r, we have to find the largest square that can be inscribed in the semicircle, with base lying on the diameter.
Examples:
Input: r = 5 Output: 20 Input: r = 8 Output: 51.2
Approach: Let r be the radius of the semicircle & a be the side length of the square.
From the figure we can see that, centre of the circle is also the midpoint of the base of the square. So in the right angled triangle AOB, from Pythagoras Theorem:
a^2 + (a/2)^2 = r^2
5*(a^2/4) = r^2
a^2 = 4*(r^2/5) i.e. area of the square
Below is the implementation of the above approach:
C++
// C++ Program to find the biggest square // which can be inscribed within the semicircle #include <bits/stdc++.h> using namespace std; // Function to find the area // of the square float squarearea( float r) { // the radius cannot be negative if (r < 0) return -1; // area of the square float a = 4 * ( pow (r, 2) / 5); return a; } // Driver code int main() { float r = 5; cout << squarearea(r) << endl; return 0; } |
Java
// Java Program to find the biggest square // which can be inscribed within the semicircle import java.io.*; class GFG { // Function to find the area // of the square static float squarearea( float r) { // the radius cannot be negative if (r < 0 ) return - 1 ; // area of the square float a = 4 * ( float )(Math.pow(r, 2 ) / 5 ); return a; } // Driver code public static void main (String[] args) { float r = 5 ; System.out.println( squarearea(r)); } } // This code is contributed by chandan_jnu. |
Python3
# Python 3 program to find the # biggest square which can be # inscribed within the semicircle # Function to find the area # of the square def squarearea(r): # the radius cannot be # negative if (r < 0 ): return - 1 # area of the square a = 4 * ( pow (r, 2 ) / 5 ) return a # Driver code if __name__ = = "__main__" : r = 5 print ( int (squarearea(r))) # This code is contributed # by ChitraNayal |
C#
// C# Program to find the // biggest square which can be // inscribed within the semicircle using System; class GFG { // Function to find the // area of the square static float squarearea( float r) { // the radius cannot be negative if (r < 0) return -1; // area of the square float a = 4 * ( float )(Math.Pow(r, 2) / 5); return a; } // Driver code public static void Main () { float r = 5; Console.WriteLine(squarearea(r)); } } // This code is contributed // by anuj_67 |
PHP
<?php // PHP Program to find the // biggest square which can be // inscribed within the semicircle // Function to find the area // of the square function squarearea( $r ) { // the radius cannot be negative if ( $r < 0) return -1; // area of the square $a = 4 * (pow( $r , 2) / 5); return $a ; } // Driver code $r = 5; echo squarearea( $r ); // This code is contributed // by Shivi_Aggarwal ?> |
Javascript
<script> // javascript Program to find the biggest square // which can be inscribed within the semicircle // Function to find the area // of the square function squarearea(r) { // the radius cannot be negative if (r < 0) return -1; // area of the square var a = 4 * (Math.pow(r, 2) / 5); return a; } // Driver code var r = 5; document.write( squarearea(r)); // This code contributed by Princi Singh </script> |
Output:
20
Time Complexity: O(1)
Auxiliary Space: O(1)