Count number of triangles possible with length of sides not exceeding N

Given an integer N, the task is to find the total number of right angled triangles that can be formed such that the length of any side of the triangle is at most N.

A right-angled triangle satisfies the following condition: X² + Y² = Z²where Z represents the length of the hypotenuse, and X and Y represent the lengths of the remaining two sides.

Examples:

Input: N = 5
Output: 1
Explanation:
The only possible combination of sides which form a right-angled triangle is {3, 4, 5}.

Input: N = 10
Output: 2
Explanation:
Possible combinations of sides which form a right-angled triangle are {3, 4, 5} and {6, 8, 10}.

Naive Approach: The idea is to generate every possible combination of triplets with integers from the range [1, N] and for each such combination, check whether it is a right-angled triangle or not.

Below is the implementation of the above approach:

C++

// C++ implementation of 
// the above approach
 
#include<bits/stdc++.h>

using namespace std;
 
// Function to count total
// number of right angled triangle

int right_angled(int n)
{

    // Initialise count with 0

    int count = 0;
 
    // Run three nested loops and

    // check all combinations of sides

    for (int z = 1; z <= n; z++) {

        for (int y = 1; y <= z; y++) {

            for (int x = 1; x <= y; x++) {
 
                // Condition for right

                // angled triangle

                if ((x * x) + (y * y) == (z * z)) {
 
                    // Increment count

                    count++;

                }

            }

        }

    }
 
    return count;
}
 
// Driver Code

int main()
{

    // Given N

    int n = 5;
 
    // Function Call

    cout << right_angled(n);

    return 0;
}

Java

// Java implementation of  
// the above approach 

import java.io.*;
 
class GFG{

// Function to count total
// number of right angled triangle

static int right_angled(int n)
{

    // Initialise count with 0

    int count = 0;

    // Run three nested loops and

    // check all combinations of sides

    for(int z = 1; z <= n; z++)

    {

        for(int y = 1; y <= z; y++)

        {

            for(int x = 1; x <= y; x++)

            {

                // Condition for right

                // angled triangle

                if ((x * x) + (y * y) == (z * z))

                {

                    // Increment count

                    count++;

                }

            }

        }

    }

    return count;
}
 
// Driver code

public static void main (String[] args)
{

    // Given N

    int n = 5;
 
    // Function call

    System.out.println(right_angled(n));
}
}
 
// This code is contributed by code_hunt

Python3

# Python implementation of  
# the above approach 
 
# Function to count total
# number of right angled triangle

def right_angled(n):

    # Initialise count with 0

    count = 0
 
    # Run three nested loops and

    # check all combinations of sides

    for z in range(1, n + 1):

        for y in range(1, z + 1):

            for x in range(1, y + 1):
 
                # Condition for right

                # angled triangle

                if ((x * x) + (y * y) == (z * z)):
 
                    # Increment count

                    count += 1

    return count
 
# Driver Code
 
# Given N

n = 5
 
# Function call

print(right_angled(n))
 
# This code is contributed by code_hunt

// C# implementation of 
// the above approach

using System;
 
class GFG{

// Function to count total
// number of right angled triangle

static int right_angled(int n)
{

    // Initialise count with 0

    int count = 0;

    // Run three nested loops and

    // check all combinations of sides

    for(int z = 1; z <= n; z++)

    {

        for(int y = 1; y <= z; y++)

        {

            for(int x = 1; x <= y; x++)

            {

                // Condition for right

                // angled triangle

                if ((x * x) + (y * y) == (z * z))

                {
 
                    // Increment count

                    count++;

                }

            }

        }

    }

    return count;
}

// Driver Code

public static void Main(string[] args)
{

    // Given N

    int n = 5;
 
    // Function call

    Console.Write(right_angled(n));
}
}
 
// This code is contributed by rutvik_56

Javascript

<script>
// javascript implementation of  
// the above approach 

// Function to count total
// number of right angled triangle

function right_angled(n)
{

    // Initialise count with 0

    var count = 0;

    // Run three nested loops and

    // check all combinations of sides

    for(z = 1; z <= n; z++)

    {

        for(y = 1; y <= z; y++)

        {

            for(x = 1; x <= y; x++)

            {

                // Condition for right

                // angled triangle

                if ((x * x) + (y * y) == (z * z))

                {

                    // Increment count

                    count++;

                }

            }

        }

    }

    return count;
}
 
// Driver code
//Given N

var n = 5;
 
// Function call
document.write(right_angled(n));
 
// This code is contributed by Amit Katiyar 
</script>

Output:

Time complexity: O(N³)
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized based on the idea that the third side of the triangle can be found out, if the two sides of the triangles are known. Follow the steps below to solve the problem:

Iterate up to N and generate pairs of possible length of two sides and find the third side using the relation x²+ y² = z²
If sqrt(x²+y²) is found to be an integer, store the three concerned integers in a Set in sorted order, as they can form a right angled triangle.
Print the final size of the set as the required count.

Below is the implementation of the above approach:

C++

// C++ implementation of the 
// above approach 
#include <bits/stdc++.h> 

using namespace std; 
 
// Function to count total 
// number of right angled triangle 

int right_angled(int n) 
{ 

    // Consider a set to store 

    // the three sides 

    set<pair<int, pair<int, int> > > s; 
 
    // Find possible third side 

    for (int x = 1; x <= n; x++) { 

        for (int y = 1; y <= n; y++) { 
 
            // Condition for a right 

            // angled triangle 

            if (x * x + y * y <= n * n) { 

                int z = sqrt(x * x + y * y); 
 
                // Check if the third side 

                // is an integer 

                if (z * z != (x * x + y * y)) 

                    continue; 
 
                vector<int> v; 
 
                // Push the three sides 

                v.push_back(x); 

                v.push_back(y); 

                v.push_back(sqrt(x * x + y * y)); 
 
                sort(v.begin(), v.end()); 
 
                // Insert the three sides in 

                // the set to find unique triangles 

                s.insert({ v[0], { v[1], v[2] } }); 

            } 

            else

                break; 

        } 

    } 
 
    // return the size of set 

    return s.size(); 
} 
 
// Driver code 

int main() 
{ 

    // Given N 

    int n = 5; 
 
    // Function Call 

    cout << right_angled(n); 

    return 0; 
}

Java

// Java implementation of the 
// above approach 

import java.util.*;
 
class Pair<F, S> 
{

    // First member of pair

    private F first;

    // Second member of pair

    private S second; 
 
    public Pair(F first, S second) 

    {

        this.first = first;

        this.second = second;

    }
}
 
class GFG{
 
// Function to count total 
// number of right angled triangle     

public static int right_angled(int n)
{

    // Consider a set to store

    // the three sides

    Set<Pair<Integer, 

        Pair<Integer,

             Integer>>> s = new HashSet<Pair<Integer,

                                        Pair<Integer,

                                             Integer>>>();

    // Find possible third side

    for(int x = 1; x <= n; x++) 

    {

        for(int y = 1; y <= n; y++)

        {

            // Condition for a right

            // angled triangle

            if (x * x + y * y <= n * n)

            {

                int z = (int)Math.sqrt(x * x + y * y);

                // Check if the third side

                // is an integer

                if (z * z != (x * x + y * y))

                    continue;

                Vector<Integer> v = new Vector<Integer>();

                // Push the three sides

                v.add(x);

                v.add(y);

                v.add((int)Math.sqrt(x * x + y * y));

                Collections.sort(v);

                // Add the three sides in

                // the set to find unique triangles

                s.add(new Pair<Integer, 

                          Pair<Integer,

                               Integer>>(v.get(0),

                      new Pair<Integer,

                               Integer>(v.get(1), 

                                        v.get(2))));

            }

            else

                break;

        }

    }

    // Return the size of set

    return s.size() - 1;
}

// Driver code

public static void main(String[] args)
{

    // Given N

    int n = 5;

    // Function call

    System.out.println(right_angled(n));
}
}
 
// This code is contributed by grand_master

Python3

# Python implementation of the
# above approach

import math
 
# Function to count total
# number of right angled triangle
 
def  right_angled(n):

    # Consider a set to store

    # the three sides

    s={}

    # Find possible third side

    for x in range(1,n+1):

        for y in range(1,n+1):

            # Condition for a right

            # angled triangle

            if(x*x+y*y<=n*n):

                z=int(math.sqrt(x*x+y*y))

                # Check if the third side

                # is an integer

                if (z*z!=(x*x+y*y)):

                    continue

                v=[]

                # Push the three sides

                v.append(x)

                v.append(y)

                v.append(int(math.sqrt(x*x+y*y)))

                v.sort()

                # Insert the three sides in

                # the set to find unique triangles

                s[v[0]]=[v[1],v[2]]

            else:

                break

    # return the size of set

    return len(s)

# Driver code
 
# Given N

n=5
 
# Function Call

print(right_angled(n))
 
# This code is contributed by Aman Kumar.

Javascript

// JavaScript implementation of the 
// above approach 
 
// Function to count total 
// number of right angled triangle 

function right_angled(n) 
{ 

    // Consider a set to store 

    // the three sides 

    let s = new Set();
 
    // Find possible third side 

    for (let x = 1; x <= n; x++) { 

        for (let y = 1; y <= n; y++) { 
 
            // Condition for a right 

            // angled triangle 

            if (x * x + y * y <= n * n) { 

                let z = Math.floor(Math.sqrt(x * x + y * y)); 
 
                // Check if the third side 

                // is an integer 

                if (z * z != (x * x + y * y)) 

                    continue; 
 
                let v = new Array(); 
 
                // Push the three sides 

                v.push(x); 

                v.push(y); 

                v.push(Math.floor(Math.sqrt(x * x + y * y))); 
 
                v.sort();
 
                // Insert the three sides in 

                // the set to find unique triangles 

                s.add([v[0],[v[1], v[2]]].join()); 

            } 

            else

                break; 

        } 

    } 
 
    // return the size of set 

    return s.size; 
} 
 
// Driver code 
// Given N 
let n = 5; 
 
// Function Call 
console.log(right_angled(n));
 
// The code is contributed by Gautam goel (gautamgoel962)

// C# implementation of the
// above approach

using System;

using System.Collections.Generic;
 
class Pair<F, S> {
 
    // First member of pair

    private F first;
 
    // Second member of pair

    private S second;
 
    public Pair(F first, S second)

    {

        this.first = first;

        this.second = second;

    }
}
 
class GFG {
 
    // Function to count total

    // number of right angled triangle

    public static int right_angled(int n)

    {
 
        // Consider a set to store

        // the three sides

        HashSet<Pair<int, Pair<int, int> > > s

            = new HashSet<Pair<int, Pair<int, int> > >();
 
        // Find possible third side

        for (int x = 1; x <= n; x++) {

            for (int y = 1; y <= n; y++) {
 
                // Condition for a right

                // angled triangle

                if (x * x + y * y <= n * n) {

                    int z = (int)Math.Sqrt(x * x + y * y);
 
                    // Check if the third side

                    // is an integer

                    if (z * z != (x * x + y * y))

                        continue;
 
                    List<int> v = new List<int>();
 
                    // Push the three sides

                    v.Add(x);

                    v.Add(y);

                    v.Add((int)Math.Sqrt(x * x + y * y));
 
                    v.Sort();
 
                    // Add the three sides in

                    // the set to find unique triangles

                    s.Add(new Pair<int, Pair<int, int> >(

                        v[0],

                        new Pair<int, int>(v[1], v[2])));

                }

                else

                    break;

            }

        }
 
        // Return the size of set

        return s.Count - 1;

    }
 
    // Driver code

    public static void Main(string[] args)

    {
 
        // Given N

        int n = 5;
 
        // Function call

        Console.WriteLine(right_angled(n));

    }
}
 
// This code is contributed by phasing17