Given an integer L, representing the side length of an N-sided regular polygon and integer K, the task is to find the side length of the Kth N-sided regular polygon formed inside the (K – 1)th regular polygon by connecting midpoints of the sides of the (K – 1)th polygon.
Examples:
Input: N = 3, L = 6, K = 2
Output: 3Input: N = 5, L = 21, K = 7
Output: 5.88796
Approach: The given problem can be solved based on the following observations:
- Suppose, \theta represents the interior angle of the N-sided polygon which is the same for all the polygons formed inside i.e.,
- The length of the side of the first polygon formed inside by connecting midpoints of the sides can be calculated using the formula as
. - The length of the side of the Kth polygon formed inside the (K – 1)th polygon and connecting midpoints of the sides of the (K – 1)th polygon is
.
Follow the steps below to solve the problem:
- Find the interior angle of the N-sided regular polygon and store it in a variable say angle in radians.
- Print the side length after calculating the side length of Kth N sided regular polygon by the above-discussed formula .
.Below is the implementation of the above approach:
C++
// C++ program for the above approach#include <bits/stdc++.h>using namespace std;#define PI 3.14159265// Function to calculate the interior// angle of a N-sided regular polygondouble findInteriorAngle(int n){ return (n - 2) * PI / n;}// Function to find the K-th polygon// formed inside the (K - 1)th polygondouble calculateSideLength(double L, int N, int K){ // Stores the interior angle double angle = findInteriorAngle(N); // Stores the side length of // K-th regular polygon double length = L * pow(sin(angle / 2), (K - 1)); // Return the length return length;}// Driver Codeint main(){ double N = 5, L = 21, K = 7; cout << calculateSideLength(L, N, K); return 0;} |
Java
// Java program for the above approachimport java.util.*;class GFG{ static final double PI = 3.14159265;// Function to calculate the interior// angle of a N-sided regular polygonstatic double findInteriorAngle(int n){ return ((n - 2) * PI) / n;}// Function to find the K-th polygon// formed inside the (K - 1)th polygonstatic double calculateSideLength(double L, int N, int K){ // Stores the interior angle double angle = findInteriorAngle(N); // Stores the side length of // K-th regular polygon double length = L * Math.pow(Math.sin(angle / 2), (K - 1)); // Return the length return length;}// Driver Codepublic static void main(String[] args){ double L = 21; int N = 5, K = 7; System.out.print(calculateSideLength(L, N, K));}}// This code is contributed by 29AjayKumar |
Python3
# Python3 program for the above approachimport mathPI = 3.14159265# Function to calculate the interior# angle of a N-sided regular polygondef findInteriorAngle(n): return (n - 2) * PI / n# Function to find the K-th polygon# formed inside the (K - 1)th polygondef calculateSideLength(L, N, K): # Stores the interior angle angle = findInteriorAngle(N) # Stores the side length of # K-th regular polygon length = L * pow(math.sin(angle / 2), (K - 1)) # Return the length return length# Driver Codeif __name__ == "__main__": N = 5 L = 21 K = 7 print(calculateSideLength(L, N, K)) # This code is contributed by ukasp. |
C#
// C# program for the above approachusing System;class GFG{ static readonly double PI = 3.14159265;// Function to calculate the interior// angle of a N-sided regular polygonstatic double findInteriorAngle(int n){ return ((n - 2) * PI) / n;}// Function to find the K-th polygon// formed inside the (K - 1)th polygonstatic double calculateSideLength(double L, int N, int K){ // Stores the interior angle double angle = findInteriorAngle(N); // Stores the side length of // K-th regular polygon double length = L * Math.Pow(Math.Sin(angle / 2), (K - 1)); // Return the length return length;}// Driver Codepublic static void Main(String[] args){ double L = 21; int N = 5, K = 7; Console.Write(calculateSideLength(L, N, K));}}// This code is contributed by 29AjayKumar |
Javascript
<script> // Javascript program for the above approach const PI = 3.14159265 // Function to calculate the interior // angle of a N-sided regular polygon function findInteriorAngle(n) { return (n - 2) * PI / n; } // Function to find the K-th polygon // formed inside the (K - 1)th polygon function calculateSideLength(L, N, K) { // Stores the interior angle let angle = findInteriorAngle(N); // Stores the side length of // K-th regular polygon let length = L * Math.pow(Math.sin(angle / 2), (K - 1)); // Return the length return length; } // Driver Code let N = 5 let L = 21 let K = 7; document.write(calculateSideLength(L, N, K)) </script> |
Output:
5.88796
Time Complexity: O(log K)
Auxiliary Space: O(1), since no extra space has been taken.
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