Given sphere of radius . 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 = 10Output: r = 9.42809, h = 13.3333Input: R = 25Output: 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) and h=x+R**

Now using these values we get,

To maximize the volume of the cone(V):

V = (πr^{2}h)/3

From the diagram,V = (πr^{2}R)/3 + πr^{2}√(R^{2}– r^{2})/3

Taking first derivative ofVwith respect torwe get,

Now, settingdV/dr = 0we get,

Squaring both sides and solving we get,

since,h = R + √(R^{2}– r^{2})

Now calculating the second derivative we get*** QuickLaTeX cannot compile formula: *** Error message: Error: Nothing to show, formula is emptyThus r=(2R√2)/3 is point of maxima

So,h = 4R/3

Below is the implementation of the above approach:

## C++

`// 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;` `}` |

## Java

`// 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` |

## Python3

`# 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` |

## C#

`// 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` |

## PHP

`<?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` `?>` |

## Javascript

`<script>` `// javascript 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` ` ` `var` `r = (2 *` ` ` `Math.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` ` ` `var` `h = (4 * R) / 3;` ` ` `return` `h;` `}` `// Driver code` `var` `R = 10;` `document.write(` `"r = "` `+ coner(R).toFixed(5) +` ` ` `", "` `+ ` `"h = "` `+ coneh(R).toFixed(5));` `// This code is contributed by 29AjayKumar` `</script>` |

**Output:**

r = 9.42809, h = 13.3333

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