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Code Geometric

Find surface area of dodecahedron

A dodecahedron is a three-dimensional polyhedron with twelve regular pentagonal faces. Finding the surface area of a dodecahedron involves calculating the total area of all its pentagonal faces. This calculation is important in geometry, crystallography, and architectural design, where dodecahedra appear in various contexts.

Problem Statement

Given the length of a side (a) of the dodecahedron, the task is to calculate its surface area. The formula for finding the surface area of a dodecahedron in terms of its side length 'a' is:

Surface Area = 3 * √(25 + 10 * √5) * a²

Example Scenario

Imagine you are a mathematician studying the properties of different geometric shapes. To understand the concept of surface area better, you decide to calculate the surface area of a dodecahedron with a side length of 7 units. This calculation helps you visualize the relationship between the number of faces and the overall area of the polyhedron.

Idea to Solve the Problem

To solve this problem, we can follow these steps:

  1. Accept the side length 'a' of the dodecahedron as an input.
  2. Use the formula to calculate the surface area of the dodecahedron.
  3. Display the calculated surface area.

Pseudocode

function dodecahedron_surface_area(a):
    area = 3 * √(25 + 10 * √5) * a²
    return area

main:
    a1 = 7
    a2 = 4
    a3 = 6.2
    
    area1 = dodecahedron_surface_area(a1)
    area2 = dodecahedron_surface_area(a2)
    area3 = dodecahedron_surface_area(a3)
    
    print("Dodecahedron [ Side :", a1, "]")
    print("Surface Area :", area1)
    
    print("Dodecahedron [ Side :", a2, "]")
    print("Surface Area :", area2)
    
    print("Dodecahedron [ Side :", a3, "]")
    print("Surface Area :", area3)

Algorithm Explanation

  1. Define a function dodecahedron_surface_area that takes the side length 'a' as an input.
  2. Inside the function, use the provided formula to calculate the surface area of the dodecahedron.
  3. In the main function, set three test cases with different side lengths 'a'.
  4. Calculate the surface areas for each test case by calling the dodecahedron_surface_area function.
  5. Display the calculated surface areas along with their respective side length values.

Code Solution

/*
  C Program 
  Find surface area of dodecahedron
*/
#include <stdio.h>

#include <math.h>

//Calculate surface area of dodecahedron by side
void surface_area(double side)
{
  // Formula of dodecahedron surface area 
  // 3 √(25 + 10 √(5)) a²
  // here a is side
  printf("\nGiven Side : %lf", side);
  //Calculate surface of dodecahedron
  double area = ((3 * sqrt(25 + 10 * (sqrt(5)))) * (side * side));
  printf("\nSurface area of dodecahedron : %lf\n", area);
}
int main()
{
  //Simple Case
  surface_area(7);
  surface_area(4);
  surface_area(6.2);
  return 0;
}

Output

Given Side : 7.000000
Surface area of dodecahedron : 1011.640712

Given Side : 4.000000
Surface area of dodecahedron : 330.331661

Given Side : 6.200000
Surface area of dodecahedron : 793.621815
// Java Program
// Find surface area of dodecahedron
class Dodecahedron
{
  //Calculate surface area of dodecahedron by side
  public void surface_area(double side)
  {
    // Formula of dodecahedron surface area 
    // 3 √(25 + 10 √(5)) a²
    // here a is side
    System.out.print("\nGiven Side : " + side);
    //Calculate surface of dodecahedron
    double area = ((3 * Math.sqrt(25 + 10 * (Math.sqrt(5)))) * (Math.pow(side, 2)));
    System.out.print("\nSurface area of dodecahedron : " + area + "\n");
  }
  public static void main(String[] args)
  {
    Dodecahedron obj = new Dodecahedron();
    //Simple Case
    obj.surface_area(7);
    obj.surface_area(4);
    obj.surface_area(6.2);
  }
}

Output

Given Side : 7.0
Surface area of dodecahedron : 1011.6407115463126

Given Side : 4.0
Surface area of dodecahedron : 330.33166091308163

Given Side : 6.2
Surface area of dodecahedron : 793.6218153436787
// C++ Program
// Find surface area of dodecahedron
#include<iostream>
#include<math.h>
using namespace std;
class Dodecahedron
{
  public:
    //Calculate surface area of dodecahedron by side
    void surface_area(double side)
    {
      cout << "\nGiven Side : " << side;
      //Calculate surface of dodecahedron
      double area = ((3 * sqrt(25 + 10 * (sqrt(5)))) * (side * side));
      cout << "\nSurface area of dodecahedron : " << area << "\n";
    }
};
int main()
{
  Dodecahedron obj ;
  //Simple Case
  obj.surface_area(7);
  obj.surface_area(4);
  obj.surface_area(6.2);
  return 0;
}

Output

Given Side : 7
Surface area of dodecahedron : 1011.64

Given Side : 4
Surface area of dodecahedron : 330.332

Given Side : 6.2
Surface area of dodecahedron : 793.622
// C# Program
// Find surface area of dodecahedron
using System;
class Dodecahedron
{
  //Calculate surface area of dodecahedron by side
  public void surface_area(double side)
  {
    Console.Write("\nGiven Side : " + side);
    //Calculate surface of dodecahedron
    double area = ((3 * Math.Sqrt(25 + 10 * (Math.Sqrt(5)))) * (side * side));
    Console.Write("\nSurface area of dodecahedron : " + area + "\n");
  }
  public static void Main(String[] args)
  {
    Dodecahedron obj = new Dodecahedron();
    //Simple Case
    obj.surface_area(7);
    obj.surface_area(4);
    obj.surface_area(6.2);
  }
}

Output

Given Side : 7
Surface area of dodecahedron : 1011.64071154631

Given Side : 4
Surface area of dodecahedron : 330.331660913082

Given Side : 6.2
Surface area of dodecahedron : 793.621815343679
<?php
// Php Program
// Find surface area of dodecahedron
class Dodecahedron
{
  //Calculate surface area of dodecahedron by side
  public  function surface_area($side)
  {
    echo "\nGiven Side : ". $side;
    //Calculate surface of dodecahedron
    $area = ((3 * sqrt(25 + 10 * (sqrt(5)))) * ($side * $side));
    echo "\nSurface area of dodecahedron : ". $area ."\n";
  }
}

function main()
{
  $obj = new Dodecahedron();
  //Simple Case
  $obj->surface_area(7);
  $obj->surface_area(4);
  $obj->surface_area(6.2);
}
main();

Output

Given Side : 7
Surface area of dodecahedron : 1011.6407115463

Given Side : 4
Surface area of dodecahedron : 330.33166091308

Given Side : 6.2
Surface area of dodecahedron : 793.62181534368
// Node Js Program
// Find surface area of dodecahedron
class Dodecahedron
{
  //Calculate surface area of dodecahedron by side
  surface_area(side)
  {
    process.stdout.write("\nGiven Side : " + side);
    //Calculate surface of dodecahedron
    var area = ((3 * Math.sqrt(25 + 10 * (Math.sqrt(5)))) * (side * side));
    process.stdout.write("\nSurface area of dodecahedron : " + area + "\n");
  }
}

function main()
{
  var obj = new Dodecahedron();
  //Simple Case
  obj.surface_area(7);
  obj.surface_area(4);
  obj.surface_area(6.2);
}
main();

Output

Given Side : 7
Surface area of dodecahedron : 1011.6407115463126

Given Side : 4
Surface area of dodecahedron : 330.33166091308163

Given Side : 6.2
Surface area of dodecahedron : 793.6218153436787
#  Python 3 Program
#  Find surface area of dodecahedron

import math

class Dodecahedron :
  # Calculate surface area of dodecahedron by side
  def surface_area(self, side) :
    print("\nGiven Side : ", side, end = "")
    # Calculate surface of dodecahedron
    area = ((3 * math.sqrt(25 + 10 * (math.sqrt(5)))) * (side * side))
    print("\nSurface area of dodecahedron : ", area ,"\n", end = "")
  

def main() :
  obj = Dodecahedron()
  # Simple Case
  obj.surface_area(7)
  obj.surface_area(4)
  obj.surface_area(6.2)

if __name__ == "__main__": main()

Output

Given Side :  7
Surface area of dodecahedron :  1011.6407115463126

Given Side :  4
Surface area of dodecahedron :  330.33166091308163

Given Side :  6.2
Surface area of dodecahedron :  793.6218153436787
#  Ruby Program
#  Find surface area of dodecahedron
class Dodecahedron

  # Calculate surface area of dodecahedron by side
  def surface_area(side)
  
    #  Formula of dodecahedron surface area 
    #  3 √(25 + 10 √(5)) a²
    #  here a is side
    print("\nGiven Side : ", side)
    # Calculate surface of dodecahedron
    area = ((3 * Math.sqrt(25 + 10 * (Math.sqrt(5)))) * (side * side))
    print("\nSurface area of dodecahedron : ", area ,"\n")
  end
end
def main()

  obj = Dodecahedron.new()
  # Simple Case
  obj.surface_area(7)
  obj.surface_area(4)
  obj.surface_area(6.2)
end
main()

Output

Given Side : 7
Surface area of dodecahedron : 1011.6407115463126

Given Side : 4
Surface area of dodecahedron : 330.33166091308163

Given Side : 6.2
Surface area of dodecahedron : 793.6218153436787

// Scala Program
// Find surface area of dodecahedron
class Dodecahedron
{
  //Calculate surface area of dodecahedron by side
  def surface_area(side: Double): Unit = {
    // Formula of dodecahedron surface area 
    // 3 √(25 + 10 √(5)) a²
    // here a is side
    print("\nGiven Side : " + side);
    //Calculate surface of dodecahedron
    var area: Double = ((3 * Math.sqrt(25 + 10 * (Math.sqrt(5)))) * (side * side));
    print("\nSurface area of dodecahedron : " + area + "\n");
  }
}
object Main
{
  def main(args: Array[String]): Unit = {
    var obj: Dodecahedron = new Dodecahedron();
    //Simple Case
    obj.surface_area(7);
    obj.surface_area(4);
    obj.surface_area(6.2);
  }
}

Output

Given Side : 7.0
Surface area of dodecahedron : 1011.6407115463126

Given Side : 4.0
Surface area of dodecahedron : 330.33166091308163

Given Side : 6.2
Surface area of dodecahedron : 793.6218153436787
// Swift Program
// Find surface area of dodecahedron
import Foundation
class Dodecahedron
{
  //Calculate surface area of dodecahedron by side
  func surface_area(_ side: Double)
  {

    // Formula of dodecahedron surface area 
    // 3 √(25 + 10 √(5)) a²
    // here a is side
    print("\nGiven Side : ", side, terminator: "");
    //Calculate surface of dodecahedron
    let area: Double = ((3 * sqrt(25 + 10 * (sqrt(5)))) * (side * side));
    print("\nSurface area of dodecahedron : ", area ,"\n", terminator: "");
  }
}
func main()
{
  let obj: Dodecahedron = Dodecahedron();
  //Simple Case
  obj.surface_area(7);
  obj.surface_area(4);
  obj.surface_area(6.2);
}
main();

Output

Given Side :  7.0
Surface area of dodecahedron :  1011.64071154631

Given Side :  4.0
Surface area of dodecahedron :  330.331660913082

Given Side :  6.2
Surface area of dodecahedron :  793.621815343679

Output Explanation

The code calculates the surface area for each test case and displays the results. For a dodecahedron with a side length 'a' = 7, the surface area is approximately 1011.640712. Similarly, for side lengths 'a' = 4 and 'a' = 6.2, the surface areas are approximately 330.331661 and 793.621815, respectively.

Time Complexity

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

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