Program to find Area of Triangle inscribed in N-sided Regular Polygon

Given the triangle inscribed in an N-sided regular polygon with given side length, formed using any 3 vertices of the polygon, the task is to find the area of this triangle.

Examples:

Input: N = 6, side = 10
Output: 129.904

Input: N = 8, side = 5
Output: 45.2665

Approach: Consider the 1st example:



  • Given is a 6 sided regular polygon ABCDEF with a triangle AEC inscribed in it.
  • As it can be seen, the triangle divides given polygon into 6 equal triangular areas, where the point of intersection of triangle AEC is the centroid of the triangle.

  • Find the area of the regular polygon. Area of the regular polygon can be calculated with the help of formula (A*P)/2 where P is the perimeter of that polygon and A is apothem of that polygon.
  • Area of each of the triangulated part will be (TriangulatedArea = Area of N sided regular polygon / N) from the law of symmetry.
  • Since the Triangle ACE comprises of 3 out of 6 in it, So the area of triangle ACE will be (3 * TriangulatedArea)
  • Therefore, in general, if there is an N-sided regular polygon with area A, the area of a triangle inscribed in it will be (A/N)*3.
  • Below is the implementation of the above approach:

    C++

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    // C++ Program to find the area of a triangle
    // inscribed in N-sided regular polygon
      
    #include <bits/stdc++.h>
    #include <cmath>
    using namespace std;
      
    // Function to find the area of the polygon
    double area_of_regular_polygon(double n, double len)
    {
      
        // area of a regular polygon with N sides
        // and side length len
        double P = (len * n);
        double A
            = len
              / (2 * tan((180 / n)
                         * 3.14159 / 180));
        double area = (P * A) / 2;
      
        return area;
    }
      
    // Function to find the area of a triangle
    double area_of_triangle_inscribed(double n, double len)
    {
      
        double area = area_of_regular_polygon(n, len);
      
        // area of one triangle
        // in an N-sided regular polygon
        double triangle = area / n;
      
        // area of inscribed triangle
        double ins_tri = (triangle * 3);
      
        return ins_tri;
    }
      
    // Driver code
    int main()
    {
        double n = 6, len = 10;
      
        cout << area_of_triangle_inscribed(n, len)
             << endl;
      
        return 0;
    }

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    Java

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    // Java Program to find the area of a triangle
    // inscribed in N-sided regular polygon
    import java.util.*;
      
    class GFG
    {
      
    // Function to find the area of the polygon
    static double area_of_regular_polygon(double n, 
                                          double len)
    {
      
        // area of a regular polygon with N sides
        // and side length len
        double P = (len * n);
        double A = len / (2 * Math.tan((180 / n) * 
                                 3.14159 / 180));
        double area = (P * A) / 2;
      
        return area;
    }
      
    // Function to find the area of a triangle
    static double area_of_triangle_inscribed(double n, 
                                             double len)
    {
        double area = area_of_regular_polygon(n, len);
      
        // area of one triangle
        // in an N-sided regular polygon
        double triangle = area / n;
      
        // area of inscribed triangle
        double ins_tri = (triangle * 3);
      
        return ins_tri;
    }
      
    // Driver code
    static public void main(String[] arg) 
    {
        double n = 6, len = 10;
      
        System.out.printf("%.3f"
               area_of_triangle_inscribed(n, len));
    }
    }
      
    // This code is contributed by PrinciRaj1992

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    Python3

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    # Python3 Program to find the area 
    # of a triangle inscribed in 
    # N-sided regular polygon 
    import math 
      
    # Function to find the area of the polygon 
    def area_of_regular_polygon(n, len): 
      
        # area of a regular polygon with 
        # N sides and side length len 
        P = (len * n); 
        A = len / (2 * math.tan((180 / n) * 
                          3.14159 / 180)) 
        area = (P * A) / 2
      
        return area 
      
    # Function to find the area of a triangle 
    def area_of_triangle_inscribed(n, len): 
      
        area = area_of_regular_polygon(n, len
      
        # area of one triangle 
        # in an N-sided regular polygon 
        triangle = area /
      
        # area of inscribed triangle 
        ins_tri = (triangle * 3); 
      
        return ins_tri 
      
    # Driver code 
    n = 6
    len = 10
    print(round(area_of_triangle_inscribed(n, len), 3)) 
      
    # This code is contributed by divyamohan

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    C#

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    // C# Program to find the area of a triangle
    // inscribed in N-sided regular polygon
    using System;
                          
    class GFG
    {
      
    // Function to find the area of the polygon
    static double area_of_regular_polygon(double n, 
                                          double len)
    {
      
        // area of a regular polygon with N sides
        // and side length len
        double P = (len * n);
        double A = len / (2 * Math.Tan((180 / n) * 
                                 3.14159 / 180));
        double area = (P * A) / 2;
      
        return area;
    }
      
    // Function to find the area of a triangle
    static double area_of_triangle_inscribed(double n, 
                                             double len)
    {
        double area = area_of_regular_polygon(n, len);
      
        // area of one triangle
        // in an N-sided regular polygon
        double triangle = area / n;
      
        // area of inscribed triangle
        double ins_tri = (triangle * 3);
      
        return ins_tri;
    }
      
    // Driver code
    static public void Main(String[] arg) 
    {
        double n = 6, len = 10;
      
        Console.Write("{0:F3}"
                area_of_triangle_inscribed(n, len));
    }
    }
      
    // This code is contributed by PrinciRaj1992

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    Output:

    129.904
    


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