Find the volume of a torus

A torus is a three-dimensional geometric shape resembling a donut with a hole in the center. Finding the volume of a torus involves calculating the amount of space enclosed within its boundaries. This calculation is relevant in various fields, such as mathematics, physics, and engineering, where tori are used to model objects like pipes, tires, and certain parts of machinery.

Problem Statement

Given the radii of the small circle (r) and the large circle (R) that make up the torus, the task is to calculate its volume. The formula for finding the volume of a torus in terms of its radii 'r' and 'R' is:

Volume = 2 * π² * R * r²

Example Scenario

Imagine you are an engineer designing a water pipe system for a new building. To estimate the amount of water the pipes can hold, you need to calculate the volume of the toroidal pipe shape. This calculation helps ensure that the pipe system can handle the required water flow.

Idea to Solve the Problem

To solve this problem, we can follow these steps:

1. Accept the radii 'r' and 'R' of the small and large circles as inputs.
2. Use the formula to calculate the volume of the torus.
3. Display the calculated volume.

Pseudocode

function torus_volume(r, R):
    volume = 2 * π² * R * (r * r)
    return volume

main:
    r1 = 3
    R1 = 7
    
    r2 = 4
    R2 = 8
    
    r3 = 6.2
    R3 = 8.3
    
    volume1 = torus_volume(r1, R1)
    volume2 = torus_volume(r2, R2)
    volume3 = torus_volume(r3, R3)
    
    print("Torus [ r :", r1, "R :", R1, "]")
    print("Volume :", volume1)
    
    print("Torus [ r :", r2, "R :", R2, "]")
    print("Volume :", volume2)
    
    print("Torus [ r :", r3, "R :", R3, "]")
    print("Volume :", volume3)

Algorithm Explanation

1. Define a function torus_volume that takes the radii 'r' and 'R' as inputs.
2. Inside the function, use the provided formula to calculate the volume of the torus.
3. In the main function, set three test cases with different values for the radii 'r' and 'R'.
4. Calculate the volumes for each test case by calling the torus_volume function.
5. Display the calculated volumes along with their respective radii values.

Code Solution

• C
• Java
• C++
• C#
• Php
• Node Js
• Python
• Ruby
• Scala
• Swift 4

/*
  C Program 
  Find the volume of a torus
*/
#include <stdio.h>

#include <math.h>

//Calculate volume of a torus by given small and large radius
void torus_volume(double r, double R)
{
  // Formula of torus volume
  // 2 × π² × R × r²
  // here r is radius of small circle
  // R is radius of large circle
  //Display given inputs
  printf("\nGiven Radius of r: %lf Radius of R : %lf", r, R);
  //Calculate volume of torus
  double volume = 2 * (M_PI * M_PI) * R * (r * r);
  //Display volume
  printf("\nVolume of torus : %lf\n", volume);
}
int main()
{
  //Simple Case
  torus_volume(3, 7);
  torus_volume(4, 8);
  torus_volume(6.2, 8.3);
  return 0;
}

Given Radius of r: 3.000000 Radius of R : 7.000000
Volume of torus : 1243.570155

Given Radius of r: 4.000000 Radius of R : 8.000000
Volume of torus : 2526.618727

Given Radius of r: 6.200000 Radius of R : 8.300000
Volume of torus : 6297.834047

// Java Program
// Find the volume of a torus
class Torus
{
  //Calculate volume of a torus by given small and large radius
  public void volume(double r, double R)
  {
    // Formula of torus volume
    // 2 × π² × R × r²
    // here r is radius of small circle
    // R is radius of large circle
    System.out.print("\nGiven Radius of small r: " + r + " Radius of large R : " + R + "");
    //Calculate volume of torus
    double volume = 2 * (Math.PI * Math.PI) * R * (r * r);
    System.out.print("\nVolume of torus : " + volume + "\n");
  }
  public static void main(String[] args)
  {
    Torus torus = new Torus();
    //Simple Case
    torus.volume(3, 7);
    torus.volume(4, 8);
    torus.volume(6.2, 8.3);
  }
}

Given Radius of small r: 3.0 Radius of large R : 7.0
Volume of torus : 1243.570154537259

Given Radius of small r: 4.0 Radius of large R : 8.0
Volume of torus : 2526.6187266788756

Given Radius of small r: 6.2 Radius of large R : 8.3
Volume of torus : 6297.834046752725

// C++ Program
// Find the volume of a torus
#include<iostream>
#include<math.h>

using namespace std;
class Torus
{
  public:
    //Calculate volume of a torus by given small and large radius
    void volume(double r, double R)
    {
      cout << "\nGiven Radius of small r: " << r << " Radius of large R : " << R << "";
      //Calculate volume of torus
      double volume = 2 * (M_PI * M_PI) * R * (r * r);
      cout << "\nVolume of torus : " << volume << "\n";
    }
};
int main()
{
  Torus torus ;
  //Simple Case
  torus.volume(3, 7);
  torus.volume(4, 8);
  torus.volume(6.2, 8.3);
  return 0;
}

Given Radius of small r: 3 Radius of large R : 7
Volume of torus : 1243.57

Given Radius of small r: 4 Radius of large R : 8
Volume of torus : 2526.62

Given Radius of small r: 6.2 Radius of large R : 8.3
Volume of torus : 6297.83

// C# Program
// Find the volume of a torus
using System;
class Torus
{
  //Calculate volume of a torus by given small and large radius
  public void volume(double r, double R)
  {
    Console.Write("\nGiven Radius of small r: " + r + " Radius of large R : " + R + "");
    //Calculate volume of torus
    double volume = 2 * (Math.PI * Math.PI) * R * (r * r);
    Console.Write("\nVolume of torus : " + volume + "\n");
  }
  public static void Main(String[] args)
  {
    Torus torus = new Torus();
    //Simple Case
    torus.volume(3, 7);
    torus.volume(4, 8);
    torus.volume(6.2, 8.3);
  }
}

Given Radius of small r: 3 Radius of large R : 7
Volume of torus : 1243.57015453726

Given Radius of small r: 4 Radius of large R : 8
Volume of torus : 2526.61872667888

Given Radius of small r: 6.2 Radius of large R : 8.3
Volume of torus : 6297.83404675273

<?php
// Php Program
// Find the volume of a torus
class Torus
{
  //Calculate volume of a torus by given small and large radius
  public  function volume($r, $R)
  {
    echo "\nGiven Radius of small r: ". $r ." Radius of large R : ". $R ."";
    //Calculate volume of torus
    $volume = 2 * (M_PI * M_PI) * $R * ($r * $r);
    echo "\nVolume of torus : ". $volume ."\n";
  }
}

function main()
{
  $torus = new Torus();
  //Simple Case
  $torus->volume(3, 7);
  $torus->volume(4, 8);
  $torus->volume(6.2, 8.3);
}
main();

Given Radius of small r: 3 Radius of large R : 7
Volume of torus : 1243.5701545373

Given Radius of small r: 4 Radius of large R : 8
Volume of torus : 2526.6187266789

Given Radius of small r: 6.2 Radius of large R : 8.3
Volume of torus : 6297.8340467527

// Node Js Program
// Find the volume of a torus
class Torus
{
  //Calculate volume of a torus by given small and large radius
  volume(r, R)
  {
    process.stdout.write("\nGiven Radius of small r: " + r + " Radius of large R : " + R + "");
    //Calculate volume of torus
    var volume = 2 * (Math.PI * Math.PI) * R * (r * r);
    process.stdout.write("\nVolume of torus : " + volume + "\n");
  }
}

function main()
{
  var torus = new Torus();
  //Simple Case
  torus.volume(3, 7);
  torus.volume(4, 8);
  torus.volume(6.2, 8.3);
}
main();

Given Radius of small r: 3 Radius of large R : 7
Volume of torus : 1243.570154537259

Given Radius of small r: 4 Radius of large R : 8
Volume of torus : 2526.6187266788756

Given Radius of small r: 6.2 Radius of large R : 8.3
Volume of torus : 6297.834046752725

#  Python 3 Program
#  Find the volume of a torus
import math
class Torus :
  # Calculate volume of a torus by given small and large radius
  def volume(self, r, R) :
    print("\nGiven Radius of small r: ", r ," Radius of large R : ", R ,"", end = "")
    # Calculate volume of torus
    volume = 2 * (math.pi * math.pi) * R * (r * r)
    print("\nVolume of torus : ", volume ,"\n", end = "")
  

def main() :
  torus = Torus()
  # Simple Case
  torus.volume(3, 7)
  torus.volume(4, 8)
  torus.volume(6.2, 8.3)

if __name__ == "__main__": main()

Given Radius of small r:  3  Radius of large R :  7
Volume of torus :  1243.570154537259

Given Radius of small r:  4  Radius of large R :  8
Volume of torus :  2526.6187266788756

Given Radius of small r:  6.2  Radius of large R :  8.3
Volume of torus :  6297.834046752725

#  Ruby Program
#  Find the volume of a torus
class Torus

  # Calculate volume of a torus by given small and large radius
  def volume(s_r, l_r)
  
    #  Formula of torus volume
    #  2 × π² × R × r²
    #  here r is radius of small circle
    #  R is radius of large circle
    print("\nGiven Radius of small r: ", s_r ," Radius of large R : ", l_r ,"")
    # Calculate volume of torus
    volume = 2 * (Math::PI * Math::PI) * l_r * (s_r * s_r)
    print("\nVolume of torus : ", volume ,"\n")
  end
end
def main()

  torus = Torus.new()
  # Simple Case
  torus.volume(3, 7)
  torus.volume(4, 8)
  torus.volume(6.2, 8.3)
end
main()

Given Radius of small r: 3 Radius of large R : 7
Volume of torus : 1243.570154537259

Given Radius of small r: 4 Radius of large R : 8
Volume of torus : 2526.6187266788756

Given Radius of small r: 6.2 Radius of large R : 8.3
Volume of torus : 6297.834046752725

// Scala Program
// Find the volume of a torus
class Torus
{
  //Calculate volume of a torus by given small and large radius
  def volume(r: Double, R: Double): Unit = {
    // Formula of torus volume
    // 2 × π² × R × r²
    // here r is radius of small circle
    // R is radius of large circle
    print("\nGiven Radius of small r: " + r + " Radius of large R : " + R + "");
    //Calculate volume of torus
    var volume: Double = 2 * (Math.PI * Math.PI) * R * (r * r);
    print("\nVolume of torus : " + volume + "\n");
  }
}
object Main
{
  def main(args: Array[String]): Unit = {
    var torus: Torus = new Torus();
    //Simple Case
    torus.volume(3, 7);
    torus.volume(4, 8);
    torus.volume(6.2, 8.3);
  }
}

Given Radius of small r: 3.0 Radius of large R : 7.0
Volume of torus : 1243.570154537259

Given Radius of small r: 4.0 Radius of large R : 8.0
Volume of torus : 2526.6187266788756

Given Radius of small r: 6.2 Radius of large R : 8.3
Volume of torus : 6297.834046752725

// Swift Program
// Find the volume of a torus
import Foundation
class Torus
{
  //Calculate volume of a torus by given small and large radius
  func volume(_ r: Double, _ R: Double)
  {
    print("\nGiven Radius of small r: ", r ," Radius of large R : ", R ,"", terminator: "");
    //Calculate volume of torus
    let volume: Double = 2 * (Double.pi * Double.pi) * R * (r * r);
    print("\nVolume of torus : ", volume ,"\n", terminator: "");
  }
}
func main()
{
  let torus: Torus? = Torus();
  //Simple Case
  torus!.volume(3, 7);
  torus!.volume(4, 8);
  torus!.volume(6.2, 8.3);
}
main();

Given Radius of small r:  3.0  Radius of large R :  7.0
Volume of torus :  1243.57015453726

Given Radius of small r:  4.0  Radius of large R :  8.0
Volume of torus :  2526.61872667888

Given Radius of small r:  6.2  Radius of large R :  8.3
Volume of torus :  6297.83404675273

Output Explanation

The code calculates the volume for each test case and displays the results. For a torus with radii 'r' = 3 and 'R' = 7, the volume is approximately 1243.570155. Similarly, for radii 'r' = 4 and 'R' = 8, the volume is approximately 2526.618727. The third test case follows the same pattern with a different set of radii values.

Time Complexity

The time complexity of this code is constant O(1) because the calculations involve basic arithmetic operations and the value of π², which are calculated in constant time regardless of the input size. The program performs a fixed number of operations for each test case, making it efficient.