Largest sphere that can be inscribed in a right circular cylinder inscribed in a frustum

Given here is a frustum of height h, top-radius r & base-radius R, which inscribes a right circular cylinder which in turn inscribes a sphere . The task is to find the largest possible volume of this sphere.

Examples:

Input: r = 5, R = 8, h = 11
Output: 523.333

Input: r = 9, R = 14, h = 20
Output:3052.08



Approach: Let the height of the cylinder = H, radius of the sphere = x
We know, the height and radius of the cylinder inscribed within the frustum is equal to the height and top-radius of the frustum respectively(Please refer here).So the height of the cylinder = h, radius of the cylinder = r.
Also, radius of the sphere inscribed within a cylinder is equal to radius of the cylinder(Please refer here), so x = r.
So, volume of the sphere, V = 4*π*r^3/3.

Below is the implementation of the above approach:

C++

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// C++ Program to find the biggest sphere
// that can be inscribed within a right
// circular cylinder which in turn is inscribed
// within a frustum
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the biggest sphere
float sph(float r, float R, float h)
{
  
    // the radii and height cannot be negative
    if (r < 0 && R < 0 && h < 0)
        return -1;
  
    // radius of the sphere
    float x = r;
  
    // volume of the sphere
    float V = (4 * 3.14 * pow(r, 3)) / 3;
  
    return V;
}
  
// Driver code
int main()
{
    float r = 5, R = 8, h = 11;
    cout << sph(r, R, h) << endl;
  
    return 0;
}

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Java














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// Java Program to find the biggest sphere 


// that can be inscribed within a right 


// circular cylinder which in turn is inscribed 


// within a frustum 


import java.lang.Math; 


  


class gfg 


{ 


      


// Function to find the biggest sphere 


static float sph(float r, float R, float h) 


{ 


  


    // the radii and height cannot be negative 


    if (r < 0 && R < 0 && h < 0) 


        return -1; 


  


    // radius of the sphere 


    float x = r; 


  


    // volume of the sphere 


    float V = (float)(4 * 3.14f * Math.pow(r, 3)) / 3; 


  


    return V; 


} 


  


// Driver code 


public static void main(String[] args) 


{ 


    float r = 5, R = 8, h = 11; 


    System.out.println(sph(r, R, h)); 


} 


} 


  


// This Code is contributed by Code_Mech. 



















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Python3














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# Python3 Program to find the biggest sphere 


# that can be inscribed within a right 


# circular cylinder which in turn is inscribed 


# within a frustum 


import math as mt 


  


# Function to find the biggest sphere 


def sph(r, R, h): 


  


    # the radii and height cannot 


    # be negative 


    if (r < 0 and R < 0 and h < 0): 


        return -1


  


    # radius of the sphere 


    x = r 


  


    # volume of the sphere 


    V = (4 * 3.14 * pow(r, 3)) / 3


  


    return V 


  


# Driver code 


r, R, h = 5, 8, 11


print(sph(r, R, h)) 


  


# This code is contributed by  


# Mohit kumar 29 



















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C#














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// C# Program to find the biggest sphere 


// that can be inscribed within a right 


// circular cylinder which in turn is  


// inscribed within a frustum 


using System; 


  


class gfg 


{ 


      


    // Function to find the biggest sphere 


    static float sph(float r, float R, float h) 


    { 


      


        // the radii and height  


        // cannot be negative 


        if (r < 0 && R < 0 && h < 0) 


            return -1; 


      


        // radius of the sphere 


        float x = r; 


      


        // volume of the sphere 


        float V = (float)(4 * 3.14f *  


                    Math.Pow(r, 3)) / 3; 


      


        return V; 


    } 


      


    // Driver code 


    public static void Main() 


    { 


        float r = 5, R = 8, h = 11; 


        Console.WriteLine(sph(r, R, h)); 


    } 


} 


  


// This code is contributed by Ryuga 



















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PHP














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<?php 


// PHP Program to find the biggest sphere 


// that can be inscribed within a right 


// circular cylinder which in turn is  


// inscribed within a frustum Function 


// to find the biggest sphere 


  


function sph($r, $R, $h) 


{ 


  


    // the radii and height  


    // cannot be negative 


    if ($r < 0 && $R < 0 && $h < 0) 


        return -1; 


  


    // radius of the sphere 


    $x = $r; 


  


    // volume of the sphere 


    $V = (4 * 3.14 * pow($r, 3)) / 3; 


  


    return $V; 


} 


  


// Driver code 


    $r = 5; 


    $R = 8; 


    $h = 11; 


    echo sph($r, $R, $h); 


  


#This Code is contributed by ajit.. 


?> 



















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Output:


523.333






          
          
          
          

            

    

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