Volume of cube using its space diagonal

Given the length of space diagonal of a cube as d. The task is to calculate the volume occupied by the cube with the given length of space diagonal. Space diagonal is a line connecting two vertices that are not on the same face.

Cube

Examples:



Input: d = 5 
Output: Volume of Cube: 24.0563

Input: d = 10
Output: Volume of Cube: 192.45

Volume of cube whose space diagonal is given:  \sqrt{3}  \frac{d^3}{9}

Proof:

Let d = the length of diagonal |AB| and
let a = the length of each side of the cube.
Pythagorus #1 in triangle ACD:
(AC)^2=(a)^2+(a)^2       AC = \sqrt{2}a $

Pythagorus #2 in triangle ABC:
 (AB)^2=(\sqrt{2}a)^2+(a)^2       AB = \sqrt{3}a    $
Now we can solve for a in terms of d:
$$a=\frac{d}{ \sqrt{3} }$
This means that the volume V is:
$V=a^3=\frac{d^3}{3 \sqrt{3}} = \frac{\sqrt{3}d^3}{9}$$

Below is the required implementation:

C++

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// C++ program to find the volume occupied
// by Cube with given space diagonal
#include <bits/stdc++.h>
using namespace std;
  
// Function to calculate Volume
float CubeVolume(float d)
{
    float Volume;
  
    // Formula to find Volume
    Volume = (sqrt(3) * pow(d, 3)) / 9;
  
    return Volume;
}
  
// Drivers code
int main()
{
  
    // space diagonal of Cube
    float d = 5;
  
    cout << "Volume of Cube: "
         << CubeVolume(d);
  
    return 0;
}

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Java

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// Java program to find the volume occupied 
// by Cube with given space diagonal 
  
public class GFG{
      
    // Function to calculate Volume 
    static float CubeVolume(float d) 
    
        float Volume; 
      
        // Formula to find Volume 
        Volume = (float) (Math.sqrt(3) * Math.pow(d, 3)) / 9
      
        return Volume; 
    
      
    // Drivers code 
    public static void main(String []args)
    
      
        // space diagonal of Cube 
        float d = 5
      
        System.out.println("Volume of Cube: " + CubeVolume(d)); 
      
    
  
    // This code is contributed by Ryuga
    }

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Python3

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# Python 3 program to find the volume occupied
# by Cube with given space diagonal
from math import sqrt, pow
  
# Function to calculate Volume
def CubeVolume(d):
  
    # Formula to find Volume
    Volume = (sqrt(3) * pow(d, 3)) / 9
  
    return Volume
  
# Drivers code
if __name__ == '__main__':
  
    # space diagonal of Cube
    d = 5
  
    print("Volume of Cube:",'{0:.6}'
                format(CubeVolume(d)))
  
# This code is contributed
# by SURENDRA_GANGWAR

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C#

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// C# program to find the volume occupied 
// by Cube with given space diagonal 
using System;
  
public class GFG{
      
    // Function to calculate Volume 
    static float CubeVolume(float d) 
    
        float Volume; 
      
        // Formula to find Volume 
        Volume = (float) (Math.Sqrt(3) * Math.Pow(d, 3)) / 9; 
      
        return Volume; 
    
      
    // Drivers code 
    public static void Main()
    
      
        // space diagonal of Cube 
        float d = 5; 
      
        Console.WriteLine("Volume of Cube: {0:F4}" , CubeVolume(d)); 
      
    
  
    // This code is contributed by mits
    }

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PHP

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<?php
// PHP program to find the volume occupied
// by Cube with given space diagonal
  
// Function to calculate Volume
function CubeVolume($d)
{
    $Volume;
  
    // Formula to find Volume
    $Volume = (sqrt(3) * pow($d, 3)) / 9;
  
    return $Volume;
}
  
// Driver code
  
// space diagonal of Cube
$d = 5;
  
echo "Volume of Cube: ",
         CubeVolume($d);
      
// This code is contributed by akt_mit
?>

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Output:

Volume of Cube: 24.0563

Time Complexity: O(1)



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