# Minimum volume of cone that can be circumscribed about a sphere of radius R

Given a sphere of radius R, The task is to find out the minimum volume of the cone that can be circumscribed about it.

**Examples:**

Input:R = 10Output:Volume of cone = 8373.33Explanation:Radius of cone = 14.14 and Height of cone = 40, Volume of cone = So, volume = 8373.33Input:R = 4Output:Volume of cone = 535.89

**Approach:**

we have given a sphere of radius R inscribed in Cone. We need to find out the radius and height of the cone to find out the volume of the cone.

- In triangle AOE and ALC compute sin(X) i.e. For triangle AOE and for triangle ALC

- Now, From equating both we get

- Insert the value of H in Volume i.e. and for volume to be minimum .

- From the above equation we get and putting this value in H we get

- Hence, applying the formula of volume of cone and putting and we get the desired result.

## C++

`// C++ program to find the minimum` `// volume of the cone that can be ` `// circumscribed about a sphere` `// of radius R` `#include<bits/stdc++.h>` `using` `namespace` `std;` `// Function to find the volume` `// of the cone` `float` `Volume_of_cone(` `float` `R)` `{` ` ` ` ` `// r = radius of cone` ` ` `// h = height of cone` ` ` `// Volume of cone = (1 / 3) * (3.14) * (r*r) * (h)` ` ` `// we get radius of cone from the derivation` ` ` `// is root(2) times multiple of R` ` ` `// we get height of cone from the derivation` ` ` `// is 4 times multiple of R` ` ` `float` `V = (1 / 3.0) * (3.14) * (2 * ( R * R ) ) * (4 * R);` ` ` ` ` `return` `V;` `}` ` ` `// Driver code` `int` `main()` `{` ` ` `float` `R = 10.0;` ` ` `cout << Volume_of_cone(R);` `}` ` ` `// This code is contributed by Samarth` |

## Java

`// Java program to find the minimum` `// volume of the cone that can be` `// circumscribed about a sphere` `// of radius R` `import` `java.util.*;` `class` `GFG{` `// Function to find the volume` `// of the cone` `static` `double` `Volume_of_cone(` `double` `R)` `{` ` ` ` ` `// r = radius of cone` ` ` `// h = height of cone` ` ` `// Volume of cone = (1 / 3) * (3.14) * (r*r) * (h)` ` ` `// we get radius of cone from the derivation` ` ` `// is root(2) times multiple of R` ` ` `// we get height of cone from the derivation` ` ` `// is 4 times multiple of R` ` ` `double` `V = (` `double` `)((` `1` `/ ` `3.0` `) * (` `3.14` `) * (` `2` `* (R * R)) *` ` ` `(` `4` `* R));` ` ` `return` `V;` `}` ` ` `// Driver code` `public` `static` `void` `main(String[] args)` `{` ` ` `double` `R = ` `10.0` `;` ` ` `System.out.print(Volume_of_cone(R));` `}` `}` `// This code is contributed by sapnasingh4991` |

## Python3

`# Python3 program to find the minimum` `# Volume of the cone that can be circumscribed` `# about a sphere of radius R` `import` `math` `# Function to find the volume` `# of the cone` `def` `Volume_of_cone(R):` ` ` `# r = radius of cone` ` ` `# h = height of cone` ` ` `# Volume of cone = (1 / 3) * (3.14) * (r**2) * (h)` ` ` `# we get radius of cone from the derivation` ` ` `# is root(2) times multiple of R` ` ` `# we get height of cone from the derivation` ` ` `# is 4 times multiple of R` ` ` ` ` `V ` `=` `(` `1` `/` `3` `) ` `*` `(` `3.14` `) ` `*` `(` `2` `*` `( R` `*` `*` `2` `) ) ` `*` `(` `4` `*` `R)` ` ` ` ` `return` `V` ` ` `# Driver code` `if` `__name__ ` `=` `=` `"__main__"` `:` ` ` ` ` `R ` `=` `10` ` ` ` ` `print` `(Volume_of_cone(R))` ` ` |

## C#

`// C# program to find the minimum` `// volume of the cone that can be` `// circumscribed about a sphere` `// of radius R` `using` `System;` `class` `GFG{` `// Function to find the volume` `// of the cone` `static` `double` `Volume_of_cone(` `double` `R)` `{` ` ` ` ` `// r = radius of cone` ` ` `// h = height of cone` ` ` `// Volume of cone = (1 / 3) * (3.14) * (r*r) * (h)` ` ` `// we get radius of cone from the derivation` ` ` `// is root(2) times multiple of R` ` ` `// we get height of cone from the derivation` ` ` `// is 4 times multiple of R` ` ` `double` `V = (` `double` `)((1 / 3.0) * (3.14) *` ` ` `(2 * (R * R)) * (4 * R));` ` ` `return` `V;` `}` ` ` `// Driver code` `public` `static` `void` `Main()` `{` ` ` `double` `R = 10.0;` ` ` `Console.Write(Volume_of_cone(R));` `}` `}` `// This code is contributed by Nidhi_biet` |

## Javascript

`<script>` `// Javascript program to find the minimum` `// volume of the cone that can be` `// circumscribed about a sphere` `// of radius R` ` ` `// Function to find the volume` ` ` `// of the cone` ` ` `function` `Volume_of_cone( R)` ` ` `{` ` ` `// r = radius of cone` ` ` `// h = height of cone` ` ` `// Volume of cone = (1 / 3) * (3.14) * (r*r) * (h)` ` ` `// we get radius of cone from the derivation` ` ` `// is root(2) times multiple of R` ` ` `// we get height of cone from the derivation` ` ` `// is 4 times multiple of R` ` ` `let V = ((1 / 3.0) * (3.14) * (2 * (R * R)) * (4 * R));` ` ` `return` `V;` ` ` `}` ` ` `// Driver code` ` ` `let R = 10.0;` ` ` `document.write(Volume_of_cone(R));` `// This code is contributed by 29AjayKumar` `</script>` |

**Output:**

8373.333333333332

**Time complexity**: O(1) since performing constant operations

**Auxiliary space:** O(1) since using constant variables