Largest right circular cylinder that can be inscribed within a cone which is in turn inscribed within a cube

Given here is a cube of side length a, which inscribes a cone which in turn inscribes a right circular cylinder. The task is to find the largest possible volume of this cylinder.

Examples:

Input: a = 5
Output: 232.593

Input: a = 8
Output: 952.699

Approach:
From the figure, it is very clear, height of cone, H = a and radius of the cone, R = a√2, please refer Largest cone that can be inscribed within a cube.
and, radius of the cylinder, r = 2R/3 and height of the cylinder, h = 2H/3, please refer Largest right circular cylinder that can be inscribed within a cone.
So, radius of cylinder with respect to cube, r = 2a√2/3 and height of cylinder with respect to cube, h = 2a/3.
So, volume of the cylinder, V = 16πa^3/27.

Below is the implementation of the above approach:

C++

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// C++ Program to find the biggest right circular 
// cylinder that can be inscribed within a right 
// circular cone which in turn is inscribed 
// within a cube
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the biggest
// right circular cylinder
float cyl(float a)
{
  
    // side cannot be negative
    if (a < 0)
        return -1;
  
    // radius of right circular cylinder
    float r = (2 * a * sqrt(2)) / 3;
  
    // height of right circular cylinder
    float h = (2 * a) / 3;
  
    // volume of right circular cylinder
    float V = 3.14 * pow(r, 2) * h;
  
    return V;
}
  
// Driver code
int main()
{
    float a = 5;
    cout << cyl(a) << endl;
  
    return 0;
}

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Java

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// Java Program to find the biggest right circular 
// cylinder that can be inscribed within a right 
// circular cone which in turn is inscribed 
// within a cube
import java.lang.Math;
  
class cfg 
{
  
// Function to find the biggest
// right circular cylinder
static float cyl(float a)
{
  
    // side cannot be negative
    if (a < 0)
        return -1;
  
    // radius of right circular cylinder
    float r = (2 * a *(float)(Math.sqrt (2)) / 3);
  
    // height of right circular cylinder
    float h = (2 * a) / 3;
  
    // volume of right circular cylinder
    float V =(3.14f *(float)(Math.pow(r, 2) * h));
  
    return V;
}
  
// Driver code
public static void main(String[] args)
{
    float a = 5;
    System.out.println(cyl(a));
}
}
  
// This code is contributed by Mukul Singh.

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Python3

# Python3 Program to find the biggest
# right circular cylinder that can be
# inscribed within a right circular
# cone which in turn is inscribed
# within a cube
import math as mt

# Function to find the biggest
# right circular cylinder
def cyl(a):

# side cannot be negative
if (a < 0): return -1 # radius of right circular cylinder r = (2 * a * mt.sqrt(2)) / 3 # height of right circular cylinder h = (2 * a) / 3 # volume of right circular cylinder V = 3.14 * pow(r, 2) * h return V # Driver code a = 5 print(cyl(a)) # This code is contributed by # Mohit kumar 29 [tabby title="C#"]

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// C# Program to find the biggest 
// right circular cylinder that can
// be inscribed within a right circular
// cone which in turn is inscribed 
// within a cube 
using System;
  
class GFG 
  
    // Function to find the biggest 
    // right circular cylinder 
    static float cyl(float a) 
    
  
        // side cannot be negative 
        if (a < 0) 
            return -1; 
  
        // radius of right circular cylinder 
        float r = (2 * a * (float)(Math.Sqrt (2)) / 3); 
  
        // height of right circular cylinder 
        float h = (2 * a) / 3; 
  
        // volume of right circular cylinder 
        float V =(3.14f * (float)(Math.Pow(r, 2) * h)); 
        return V; 
    
  
    // Driver code 
    public static void Main() 
    
        float a = 5; 
        Console.Write(cyl(a)); 
    
  
// This code is contributed by Rajput-Ji 

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PHP

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<?php
// PHP Program to find the biggest right 
// circular cylinder that can be inscribed 
// within a right circular cone which in 
// turn is inscribed within a cube
  
// Function to find the biggest
// right circular cylinder
function cyl( $a )
{
  
    // side cannot be negative
    if ($a < 0)
        return -1;
  
    // radius of right circular cylinder
    $r = (2 * $a * sqrt(2)) / 3;
  
    // height of right circular cylinder
    $h = (2 * $a) / 3;
  
    // volume of right circular cylinder
    $V = 3.14 * pow($r, 2) * $h;
  
    return $V;
}
  
// Driver code
$a = 5;
echo cyl($a);
  
// This code is contributed by Mahadev99
?>

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Output:

232.593


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