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 = 10 Output: Volume of cone = 8373.33 Explanation: Radius of cone = 14.14 and Height of cone = 40, Volume of cone =So, volume = 8373.33Input: R = 4 Output: 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.
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In triangle AOE and ALC compute sin(X) i.e. For triangle AOEand for triangle ALC
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Now, From equating both we get
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Insert the value of H in Volume i.e.and for volume to be minimum .
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From the above equation we getand putting this value in H we get
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Hence, applying the formula of volume of cone and puttingand 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
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