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 = 10 Output : r = 9.42809, h = 13.3333 Input : R = 25 Output : 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 = (πr2h)/3From the diagram,
V = (πr2R)/3 + πr2√(R2 – r2)/3Taking first derivative of V with respect to r we get,
Now, setting dV/dr = 0 we get,
Squaring both sides and solving we get,
since, h = R + √(R2 – r2)Now calculating the second derivative we get
Thus r=(2R√2)/3 is point of maxima
So, h = 4R/3
![\frac{2\pi r}{3}[R+\sqrt(R^2-r^2)] =\frac{ \pi r^3}{3\sqrt(R^2 - r^2}](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-9538a477f7dbc124225a6b6bd4195878_l3.png "Rendered by QuickLaTeX.com")
![2[R+\sqrt(R^2-r^2)] =\frac{r^2}{\sqrt(R^2 - r^2)}](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-089bdb273c1d5e8ca5162b005c3fa071_l3.png "Rendered by QuickLaTeX.com")
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 = 10print("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 ?> |
r = 9.42809, h = 13.3333
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