Area of decagon inscribed within the circle

Given here is a regular decagon, inscribed within a circle of radius r, the task is to find the area of the decagon.

Examples:

Input: r = 5
Output: 160.144

Input: r = 8
Output: 409.969



Approach:
We know, side of the decagon within the circle, a = r√(2-2cos36)(Refer here)
So, area of the decagon,

A = 5*a^2*(√5+2√5)/2 = 5 *(r√(2-2cos36))^2*(√5+2√5)/2=(5*r^2*(3-√5)*(√5+2√5))/4

Below is the implementation of the above approach:

C++

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// C++ Program to find the area of the decagon
// inscribed within a circle
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the area of the decagon
float area(float r)
{
  
    // radius cannot be negative
    if (r < 0)
        return -1;
  
    // area of the decagon
    float area = (5 * pow(r, 2) * (3 - sqrt(5))
                  * (sqrt(5) + (2 * sqrt(5))))
                 / 4;
    return area;
}
  
// Driver code
int main()
{
    float r = 8;
    cout << area(r) << endl;
  
    return 0;
}

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Java

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// Java Program to find the area of the decagon 
// inscribed within a circle 
  
import java.io.*;
  
class GFG {
      
// Function to find the area of the decagon 
static double area(double  r) 
  
    // radius cannot be negative 
    if (r < 0
        return -1
  
    // area of the decagon 
    double  area = (5 * Math.pow(r, 2) * (3 - Math.sqrt(5)) 
                * (Math.sqrt(5) + ((2 * Math.sqrt(5))))/ 4); 
    return area; 
  
// Driver code 
      
    public static void main (String[] args) {
        double  r = 8
        System.out.println (area(r)); 
    }
//This code is contributed by ajit
}

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Python3

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# Python3 Program to find the area of
# the decagon inscribed within a circle
from math import sqrt,pow
  
# Function to find the 
# area of the decagon
def area(r):
      
    # radius cannot be negative
    if r < 0:
        return -1
  
    # area of the decagon
    area = (5 * pow(r, 2) * (3 - sqrt(5)) * 
                 (sqrt(5) + (2 * sqrt(5))))/ 4
    return area
  
# Driver code
if __name__ == '__main__':
    r = 8
    print(area(r))
  
# This code is contributed
# by Surendra_Gangwar

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C#

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// C# Program to find the area of the 
// decagon inscribed within a circle 
using System;
  
class GFG
{
          
// Function to find the area 
// of the decagon 
static double area(double r) 
  
    // radius cannot be negative 
    if (r < 0) 
        return -1; 
  
    // area of the decagon 
    double area = (5 * Math.Pow(r, 2) * 
                  (3 - Math.Sqrt(5)) * 
                      (Math.Sqrt(5) + 
                 ((2 * Math.Sqrt(5))))/ 4); 
    return area; 
  
// Driver code 
static public void Main ()
{
    double r = 8; 
    Console.WriteLine (area(r)); 
}
}
  
// This code is contributed by akt_mit

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PHP

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<?php
// PHP Program to find the area 
// of the decagon inscribed within
// a circle
  
// Function to find the area
// of the decagon
function area($r)
{
  
    // radius cannot be negative
    if ($r < 0)
        return -1;
  
    // area of the decagon
    $area = (5 * pow($r, 2) * (3 - sqrt(5)) * 
                (sqrt(5) + (2 * sqrt(5)))) / 4;
    return $area;
}
  
// Driver code
$r = 8;
echo area($r) . "\n";
  
// This code is contributed
// by Akanksha Rai(Abby_akku)
?>

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Output:

409.969


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