求给定操作形成的第 K 个 N 边多边形的长度

给定一个整数L ，代表一个N边正多边形的边长和整数K ，任务是找到^第(K – 1)^个内形成的第K^个N边正多边形的边长 通过连接第(K – 1)^个多边形的边的中点得到正多边形。

例子：

Input: N = 3, L = 6, K = 2
Output: 3

Input: N = 5, L = 21, K = 7
Output: 5.88796

方法：根据以下观察可以解决给定的问题：

• 假设，\theta 表示 N 边多边形的内角，它对于内部形成的所有多边形都是相同的，即， $\theta = \frac{(N - 2)*180}{N}$
• 通过连接边的中点在里面形成的第一个多边形的边长可以使用以下公式计算$\frac{2*L*sin(\frac{\theta}{2})}{2}$ .
• ^个多边形和（K – 1）的侧部的连接中点^个多边形是– K的一侧的（1 K）的内部形成^第多边形的长度$L*Sin^{K - 1}(\frac{\theta}{2})$

请按照以下步骤解决问题：

• 找到 N 边正多边形的内角并将其存储在一个变量中，比如以弧度表示的角度。
• 用上面讨论的公式计算出第K^个N 边正多边形的边长后打印边长$L*Sin^{K - 1}(\frac{\theta}{2})$ .

下面是上述方法的实现：

C++

// C++ program for the above approach
 
#include 
using namespace std;
#define PI 3.14159265
 
// Function to calculate the interior
// angle of a N-sided regular polygon
double findInteriorAngle(int n)
{
    return (n - 2) * PI / n;
}
 
// Function to find the K-th polygon
// formed inside the (K - 1)th polygon
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 * pow(sin(angle / 2),
                            (K - 1));
 
    // Return the length
    return length;
}
 
// Driver Code
int main()
{
    double N = 5, L = 21, K = 7;
    cout << calculateSideLength(L, N, K);
 
    return 0;
}

Java

// Java program for the above approach
import java.util.*;
 
class GFG{
     
static final double PI = 3.14159265;
 
// Function to calculate the interior
// angle of a N-sided regular polygon
static double findInteriorAngle(int n)
{
    return ((n - 2) * PI) / n;
}
 
// Function to find the K-th polygon
// formed inside the (K - 1)th polygon
static 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 Code
public 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 approach
import math
PI = 3.14159265
 
# Function to calculate the interior
# angle of a N-sided regular polygon
def findInteriorAngle(n):
 
    return (n - 2) * PI / n
 
# Function to find the K-th polygon
# formed inside the (K - 1)th polygon
def 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 Code
if __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 approach
using System;
 
class GFG{
     
static readonly double PI = 3.14159265;
 
// Function to calculate the interior
// angle of a N-sided regular polygon
static double findInteriorAngle(int n)
{
    return ((n - 2) * PI) / n;
}
 
// Function to find the K-th polygon
// formed inside the (K - 1)th polygon
static 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 Code
public 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

 
        // 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))
 
    

输出：

5.88796

时间复杂度： O(log K)
辅助空间： O(1)

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