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

  • Last Updated : 17 Mar, 2021

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

 

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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++




// 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;
}

Java




// 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.

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

C#




// 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

PHP




<?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
?>

Javascript




<script>
 
// javascript 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
    var r = (2 * a *(Math.sqrt (2)) / 3);
 
    // height of right circular cylinder
    var h = (2 * a) / 3;
 
    // volume of right circular cylinder
    var V =(3.14 *(Math.pow(r, 2) * h));
 
    return V;
}
 
// Driver code
 
var a = 5;
document.write(cyl(a).toFixed(5));
 
// This code contributed by Princi Singh
 
</script>
Output: 
232.593

 




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