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 ` |

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## 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 ` |

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## 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)) ` ` ` |

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## 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 ` |

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**Output:**

8373.333333333332

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