Largest right circular cone that can be inscribed within a sphere which is inscribed within a cube

Given a cube of side length a, which inscribes a sphere which in turn inscribes a right circular cone. The task is to find the largest possible volume of this cone.
Examples:

Input:  a = 5
Output: 58.1481

Input: a = 8
Output: 238.175

Approach: 
Let, the height of right circular cone = h. 
Radius of the cone = r
Radius of the sphere = R
We, know radius of the sphere inside the cube, r = a/2. Please refer ( Largest sphere that can be inscribed inside a cube). 
Also, height of cone inside the sphere, h = 4r/3. 
radius of cone inside the sphere, r = 2√2r/3. Please refer (Largest right circular cone that can be inscribed within a sphere). 
So, height of the cone inside the sphere which in turn is inscribed within a cube, h = 2a/3. 
Radius of the cone inside the sphere which in turn is inscribed within a cube, r = √2a/3.
Below is the implementation of the above approach: 

C++

// C++ Program to find the biggest right circular cone
// that can be inscribed within a right circular cone
// which in turn is inscribed within a cube

#include <bits/stdc++.h>
using namespace std;

// Function to find the biggest right circular cone
float cone(float a)
{

    // side cannot be negative
    if (a < 0)
        return -1;

    // radius of right circular cone
    float r = (a * sqrt(2)) / 3;

    // height of right circular cone
    float h = (2 * a) / 3;

    // volume of right circular cone
    float V = 3.14 * pow(r, 2) * h;

    return V;
}

// Driver code
int main()
{
    float a = 5;
    cout << cone(a) << endl;

    return 0;
}

Java

// Java Program to find the biggest right circular cone
// that can be inscribed within a right circular cone
// which in turn is inscribed within a cube
import java.io.*;

class GFG
{

// Function to find the biggest right circular cone
static float cone(float a)
{

    // side cannot be negative
    if (a < 0)
        return -1;

    // radius of right circular cone
    float r = (float) (a * Math.sqrt(2)) / 3;

    // height of right circular cone
    float h = (2 * a) / 3;

    // volume of right circular cone
    float V = (float)(3.14 *Math. pow(r, 2) * h);

    return V;
}

// Driver code
public static void main (String[] args)
{
    float a = 5;
    System.out.println( cone(a));
}
}

// This code is contributed by anuj_67..

Python3

# Python3 Program to find the biggest right
# circular cone that can be inscribed within
# a right circular cone which in turn is
# inscribed within a cube
import math

# Function to find the biggest
# right circular cone
def cone(a):

    # side cannot be negative
    if (a < 0):
        return -1;

    # radius of right circular cone
    r = (a * math.sqrt(2)) / 3;

    # height of right circular cone
    h = (2 * a) / 3;

    # volume of right circular cone
    V = 3.14 * math.pow(r, 2) * h;

    return V;

# Driver code
a = 5;
print(cone(a));

# This code is contributed by
# Shivi_Aggarwal

C#

// C# Program to find the biggest
// right circular cone that can be
// inscribed within a right circular cone
// which in turn is inscribed within a cube
using System;

class GFG
{

// Function to find the biggest
// right circular cone
static double cone(double a)
{

    // side cannot be negative
    if (a < 0)
        return -1;

    // radius of right circular cone
    double r = (double) (a * Math.Sqrt(2)) / 3;

    // height of right circular cone
    double h = (2 * a) / 3;

    // volume of right circular cone
    double V = (double)(3.14 * Math.Pow(r, 2) * h);

    return Math.Round(V,4);
}

// Driver code
static void Main ()
{
    double a = 5;
    Console.WriteLine(cone(a));
}
}

// This code is contributed by chandan_jnu

Javascript

<script>

// javascript Program to find the biggest right circular cone
// that can be inscribed within a right circular cone
// which in turn is inscribed within a cube

// Function to find the biggest right circular cone
function cone(a)
{

    // side cannot be negative
    if (a < 0)
        return -1;

    // radius of right circular cone
    var r =  (a * Math.sqrt(2)) / 3;

    // height of right circular cone
    var h = (2 * a) / 3;

    // volume of right circular cone
    var V = (3.14 *Math. pow(r, 2) * h);

    return V;
}

// Driver code
var a = 5;
document.write( cone(a).toFixed(5));


// This code is contributed by Amit Katiyar

</script>

PHP

<?php
// PHP Program to find the biggest right
// circular cone that can be inscribed
// within a right circular cone which in
// turn is inscribed within a cube

// Function to find the biggest
// right circular cone
function cone($a)
{

    // side cannot be negative
    if ($a < 0)
        return -1;

    // radius of right circular cone
    $r = ($a * sqrt(2)) / 3;

    // height of right circular cone
    $h = (2 * $a) / 3;

    // volume of right circular cone
    $V = 3.14 * pow($r, 2) * $h;

    return $V;
}

// Driver code
$a = 5;
echo round(cone($a), 4);

// This code is contributed by Ryuga
?>

Output

58.1481

Time Complexity: O(1)

Auxiliary Space: O(1)