Largest right circular cone that can be inscribed within a sphere

Given sphere of radius r. The task is to find the radius of base and height of the largest right circular cone that can be inscribed within it.

Examples:

Input : R = 10
Output : r = 9.42809, h = 13.3333

Input : R = 25
Output : r = 23.5702, h = 33.3333

Approach:

Let the radius of the cone = r
height of the cone = h



From the diagram it is clear that:
x = √(R^2 – r^2)
Now,

To maximize the volume of the cone(V):
V = (πr2h)/3

From the diagram,
V = (πr2R)/3 + πr2√(R2> – r2)/3

Taking derivative of V with respect to r we get,
dV/dr = 2πr(√(R2 – r2) + R)/3 – πr2√(R2 – r2)/3

Now, setting dV/dr = 0 we get,
r = 2√2R/3

since, h = R + √(R2 – r2)

So, h = 4R/3

Below is the implementation of the above approach:

C++

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// C++ Program to find the biggest cone
// that can be inscribed within a sphere
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the radius of the cone
float coner(float R)
{
  
    // radius cannot be negative
    if (R < 0)
        return -1;
  
    // radius of the cone
    float r = (2 * sqrt(2) * R) / 3;
    return r;
}
  
// Function to find the height of the cone
float coneh(float R)
{
  
    // side cannot be negative
    if (R < 0)
        return -1;
  
    // height of the cone
    float h = (4 * R) / 3;
    return h;
}
  
// Driver code
int main()
{
    float R = 10;
  
    cout << "r = " << coner(R) << ", "
         << "h = " << coneh(R) << endl;
  
    return 0;
}

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Java














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// Java Program to find the biggest cone 


// that can be inscribed within a sphere 


import java.util.*; 


import java.lang.*; 


  


class GFG 


{ 


// Function to find the radius 


// of the cone 


static float coner(float R) 


{ 


    // radius cannot be negative 


    if (R < 0) 


        return -1; 


  


    // radius of the cone 


    float r = (float)(2 *  


            Math.sqrt(2) * R) / 3; 


    return r; 


} 


  


// Function to find the  


// height of the cone 


static float coneh(float R) 


{ 


  


    // side cannot be negative 


    if (R < 0) 


        return -1; 


  


    // height of the cone 


    float h = (4 * R) / 3; 


    return h; 


} 


  


// Driver code 


public static void main(String args[]) 


{ 


    float R = 10; 


  


    System.out.println("r = " + coner(R) +  


                       ", " + "h = " + coneh(R)); 


} 


} 


  


// This code is contributed  


// by Akanksha Rai 



















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Python3














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# Python 3 Program to find the biggest cone  


# that can be inscribed within a sphere  


import math 


  


# Function to find the radius  


# of the cone  


def coner(R): 


      


    # radius cannot be negative  


    if (R < 0): 


        return -1;  


      


    # radius of the cone  


    r = (2 * math.sqrt(2) * R) / 3


    return float(r)  


  


# Function to find the height  


# of the cone  


def coneh(R): 


      


    # side cannot be negative  


    if (R < 0): 


        return -1;  


  


    # height of the cone  


    h = (4 * R) / 3


    return float(h)  


  


# Driver code  


R = 10


print("r = " , coner(R) ,  


      ", ", "h = " , coneh(R))  


  


# This code is contributed  


# by 29AjayKumar 



















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C#














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// C# Program to find the biggest cone 


// that can be inscribed within a sphere 


using System; 


  


class GFG 


{ 


// Function to find the radius 


// of the cone 


static float coner(float R) 


{ 


    // radius cannot be negative 


    if (R < 0) 


        return -1; 


  


    // radius of the cone 


    float r = (float)(2 *  


               Math.Sqrt(2) * R) / 3; 


    return r; 


} 


  


// Function to find the  


// height of the cone 


static float coneh(float R) 


{ 


  


    // side cannot be negative 


    if (R < 0) 


        return -1; 


  


    // height of the cone 


    float h = (4 * R) / 3; 


    return h; 


} 


  


// Driver code 


public static void Main() 


{ 


    float R = 10; 


  


    Console.WriteLine("r = " + coner(R) +  


                      ", " + "h = " + coneh(R)); 


} 


} 


  


// This code is contributed  


// by Akanksha Rai 



















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PHP














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<?php 


// PHP Program to find the biggest  


// cone that can be inscribed  


// within a sphere 


  


// Function to find the radius  


// of the cone 


function coner($R) 


{ 


  


    // radius cannot be negative 


    if ($R < 0) 


        return -1; 


  


    // radius of the cone 


    $r = (2 * sqrt(2) * $R) / 3; 


    return $r; 


} 


  


// Function to find the height 


// of the cone 


function coneh($R) 


{ 


  


    // side cannot be negative 


    if ($R < 0) 


        return -1; 


  


    // height of the cone 


    $h = (4 * $R) / 3; 


    return $h; 


} 


  


// Driver code 


$R = 10; 


  


echo ("r = "); 


echo coner($R); 


echo (", "); 


echo ("h = "); 


echo (coneh($R)); 


  


// This code is contributed 


// by Shivi_Aggarwal 


?> 



















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Output:


r = 9.42809, h = 13.3333







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