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 conefloat 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 codeint 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 cubeimport java.io.*;class GFG { // Function to find the biggest right circular conestatic 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 codepublic 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 cubeimport math# Function to find the biggest # right circular conedef 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 codea = 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 cubeusing System;class GFG { // Function to find the biggest // right circular conestatic 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 codestatic void Main () { double a = 5; Console.WriteLine(cone(a));}}// This code is contributed by chandan_jnu |
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?> |
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 conefunction 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 codevar a = 5;document.write( cone(a).toFixed(5));// This code is contributed by Amit Katiyar </script> |
Output:
58.1481
Time Complexity: O(1)
Auxiliary Space: O(1)
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