Find all sides of a right angled triangle from given hypotenuse and area | Set 1

16 Jun, 2022
Given hypotenuse and area of a right angle triangle, get its base and height and if any triangle with given hypotenuse and area is not possible, print not possible.Examples:  
Input  : hypotenuse = 5,    area = 6
Output : base = 3, height = 4

Input : hypotenuse = 5, area = 7 
Output : No triangle possible with above specification.
 
 
We can use a property of right angle triangle for solving this problem, which can be stated as follows,  
A right angle triangle with fixed hypotenuse attains
maximum area, when it is isosceles i.e. both height
and base becomes equal so if hypotenuse if H, then 
by pythagorean theorem,
Base2 + Height2 = H2

For maximum area both base and height should be equal, 
b2 + b2 = H2
b = sqrt(H2/2)

Above is the length of base at which triangle attains
maximum area, given area must be less than this maximum
area, otherwise no such triangle will possible.  
Now if given area is less than this maximum area, we can do a binary search for length of base, as increasing base will increases area, it is a monotonically increasing function where binary search can be applied easily. In below code, a method is written for getting area of right angle triangle, recall that for right angle triangle area is 1⁄2*base*height and height can be calculated from base and hypotenuse using pythagorean theorem.Below is the implementation of above approach: 
C++
Java
Python3
C#
PHP
Javascript
// C++ program to get right angle triangle, given// hypotenuse and area of triangle#include <bits/stdc++.h>using namespace std;   //  limit for float comparison#define eps 1e-6   // Utility method to get area of right angle triangle,// given base and hypotenusedouble getArea(double base, double hypotenuse){    double height = sqrt(hypotenuse*hypotenuse - base*base);    return 0.5 * base * height;}   // Prints base and height of triangle using hypotenuse// and area informationvoid printRightAngleTriangle(int hypotenuse, int area){    int hsquare = hypotenuse * hypotenuse;       // maximum area will be obtained when base and height    // are equal (= sqrt(h*h/2))    double sideForMaxArea = sqrt(hsquare / 2.0);    double maxArea = getArea(sideForMaxArea, hypotenuse);       // if given area itself is larger than maxArea then no    // solution is possible    if (area > maxArea)    {        cout << "Not possiblen";        return;    }       double low = 0.0;    double high = sideForMaxArea;    double base;       // binary search for base    while (abs(high - low) > eps)    {        base = (low + high) / 2.0;        if (getArea(base, hypotenuse) >= area)            high = base;        else            low = base;    }       // get height by pythagorean rule    double height = sqrt(hsquare - base*base);    cout << base << " " << height << endl;}   // Driver code to test above methodsint main(){    int hypotenuse = 5;    int area = 6;       printRightAngleTriangle(hypotenuse, area);    return 0;}
// Java program to get right angle triangle, given// hypotenuse and area of trianglepublic class GFG { // limit for float comparison    final static double eps = (double) 1e-6; // Utility method to get area of right angle triangle,// given base and hypotenuse    static double getArea(double base, double hypotenuse) {        double height = Math.sqrt(hypotenuse * hypotenuse - base * base);        return 0.5 * base * height;    } // Prints base and height of triangle using hypotenuse// and area information    static void printRightAngleTriangle(int hypotenuse, int area) {        int hsquare = hypotenuse * hypotenuse;         // maximum area will be obtained when base and height        // are equal (= sqrt(h*h/2))        double sideForMaxArea = Math.sqrt(hsquare / 2.0);        double maxArea = getArea(sideForMaxArea, hypotenuse);         // if given area itself is larger than maxArea then no        // solution is possible        if (area > maxArea) {            System.out.print("Not possible");            return;        }         double low = 0.0;        double high = sideForMaxArea;        double base = 0;         // binary search for base        while (Math.abs(high - low) > eps) {            base = (low + high) / 2.0;            if (getArea(base, hypotenuse) >= area) {                high = base;            } else {                low = base;            }        }         // get height by pythagorean rule        double height = Math.sqrt(hsquare - base * base);        System.out.println(Math.round(base) + " " + Math.round(height));    } // Driver code to test above methods    static public void main(String[] args) {        int hypotenuse = 5;        int area = 6;         printRightAngleTriangle(hypotenuse, area);    }} // This code is contributed by 29AjayKumar
# Python 3 program to get right angle triangle, given# hypotenuse and area of triangle # limit for float comparison# define eps 1e-6import math # Utility method to get area of right angle triangle,# given base and hypotenusedef getArea(base, hypotenuse):    height = math.sqrt(hypotenuse*hypotenuse - base*base);    return 0.5 * base * height # Prints base and height of triangle using hypotenuse# and area informationdef printRightAngleTriangle(hypotenuse, area):    hsquare = hypotenuse * hypotenuse     # maximum area will be obtained when base and height    # are equal (= sqrt(h*h/2))    sideForMaxArea = math.sqrt(hsquare / 2.0)    maxArea = getArea(sideForMaxArea, hypotenuse)     # if given area itself is larger than maxArea then no    # solution is possible    if (area > maxArea):        print("Not possiblen")        return         low = 0.0    high = sideForMaxArea         # binary search for base    while (abs(high - low) > 1e-6):        base = (low + high) / 2.0        if (getArea(base, hypotenuse) >= area):            high =base        else:            low = base         # get height by pythagorean rule    height = math.ceil(math.sqrt(hsquare - base*base))    base = math.floor(base)    print(base,height) # Driver code to test above methodsif __name__ == '__main__':    hypotenuse = 5    area = 6     printRightAngleTriangle(hypotenuse, area) # This code is contributed by# Surendra_Gangwar
// C# program to get right angle triangle, given// hypotenuse and area of triangle using System;public class GFG{  // limit for float comparison     static double eps = (double) 1e-6; // Utility method to get area of right angle triangle,// given base and hypotenuse    static double getArea(double base1, double hypotenuse) {        double height = Math.Sqrt(hypotenuse * hypotenuse - base1 * base1);        return 0.5 * base1 * height;    } // Prints base and height of triangle using hypotenuse// and area information    static void printRightAngleTriangle(int hypotenuse, int area) {        int hsquare = hypotenuse * hypotenuse;         // maximum area will be obtained when base and height        // are equal (= sqrt(h*h/2))        double sideForMaxArea = Math.Sqrt(hsquare / 2.0);        double maxArea = getArea(sideForMaxArea, hypotenuse);         // if given area itself is larger than maxArea then no        // solution is possible        if (area > maxArea) {            Console.Write("Not possible");            return;        }         double low = 0.0;        double high = sideForMaxArea;        double base1 = 0;         // binary search for base        while (Math.Abs(high - low) > eps) {            base1 = (low + high) / 2.0;            if (getArea(base1, hypotenuse) >= area) {                high = base1;            } else {                low = base1;            }        }         // get height by pythagorean rule        double height = Math.Sqrt(hsquare - base1 * base1);        Console.WriteLine(Math.Round(base1) + " " + Math.Round(height));    } // Driver code to test above methods    static public void Main() {        int hypotenuse = 5;        int area = 6;         printRightAngleTriangle(hypotenuse, area);    }} // This code is contributed by 29AjayKumar
<?php// PHP program to get right angle triangle,// given hypotenuse and area of triangle // limit for float comparison$eps =.0000001; // Utility method to get area of right// angle triangle, given base and hypotenusefunction getArea($base, $hypotenuse){    $height = sqrt($hypotenuse * $hypotenuse -                                 $base * $base);    return 0.5 * $base * $height;} // Prints base and height of triangle// using hypotenuse and area informationfunction printRightAngleTriangle($hypotenuse,                                 $area){    global $eps;    $hsquare = $hypotenuse * $hypotenuse;     // maximum area will be obtained when base    // and height are equal (= sqrt(h*h/2))    $sideForMaxArea = sqrt($hsquare / 2.0);    $maxArea = getArea($sideForMaxArea,                       $hypotenuse);     // if given area itself is larger than    // maxArea then no solution is possible    if ($area > $maxArea)    {        echo "Not possiblen";        return;    }     $low = 0.0;    $high = $sideForMaxArea;    $base;     // binary search for base    while (abs($high - $low) > $eps)    {        $base = ($low + $high) / 2.0;        if (getArea($base, $hypotenuse) >= $area)            $high = $base;        else            $low = $base;    }     // get height by pythagorean rule    $height = sqrt($hsquare - $base * $base);        echo (ceil($base)) ," ",             (floor($height)), "\n";} // Driver Code$hypotenuse = 5;$area = 6; printRightAngleTriangle($hypotenuse, $area); // This code is contributed by Sachin?>
<script> // JavaScript program to get right angle triangle, given// hypotenuse and area of triangle // limit for float comparison    let eps =  1e-6;   // Utility method to get area of right angle triangle,// given base and hypotenuse    function getArea(base, hypotenuse) {        let height = Math.sqrt(hypotenuse * hypotenuse - base * base);        return 0.5 * base * height;    }   // Prints base and height of triangle using hypotenuse// and area information    function printRightAngleTriangle(hypotenuse, area) {        let hsquare = hypotenuse * hypotenuse;           // maximum area will be obtained when base and height        // are equal (= sqrt(h*h/2))        let sideForMaxArea = Math.sqrt(hsquare / 2.0);        let maxArea = getArea(sideForMaxArea, hypotenuse);           // if given area itself is larger than maxArea then no        // solution is possible        if (area > maxArea) {            document.write("Not possible");            return;        }           let low = 0.0;        let high = sideForMaxArea;        let base = 0;           // binary search for base        while (Math.abs(high - low) > eps) {            base = (low + high) / 2.0;            if (getArea(base, hypotenuse) >= area) {                high = base;            } else {                low = base;            }        }           // get height by pythagorean rule        let height = Math.sqrt(hsquare - base * base);        document.write(Math.round(base) + " " + Math.round(height));    }     // Driver Code        let hypotenuse = 5;         let area = 6;           printRightAngleTriangle(hypotenuse, area); // This code is contributed by chinmoy1997pal.</script>
Output:  
3 4
Time complexity: O(1)Auxiliary Space: O(1)
One more solution is discussed in below post. Check if right angles possible from given area and hypotenuseThis article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
29AjayKumar
SURENDRA_GANGWAR
Sach_Code
chinmoy1997pal
hasani
Binary Search
triangle
Geometric
Mathematical
Mathematical
Geometric
Binary Search
Writing code in comment?
Please use ide.geeksforgeeks.org,
generate link and share the link here.
Program for distance between two points on earth
Optimum location of point to minimize total distance
Convex Hull | Set 1 (Jarvis's Algorithm or Wrapping)
Line Clipping | Set 1 (Cohen–Sutherland Algorithm)
Check whether a given point lies inside a triangle or not
Set in C++ Standard Template Library (STL)
Write a program to print all permutations of a given string
C++ Data Types
Merge two sorted arrays
Operators in C / C++