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Program to find Area of Triangle inscribed in N-sided Regular Polygon
  • Last Updated : 16 Mar, 2021
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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.
     



  •  

 

 

  • 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++




// 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;
}

Java




// 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

Python3




# 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 / n
 
    # 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

C#




// 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

Javascript




<script>
 
// javascript Program to find the area of a triangle
// inscribed in N-sided regular polygon
 
// Function to find the area of the polygon
function area_of_regular_polygon(n, len)
{
 
    // area of a regular polygon with N sides
    // and side length len
    let P = (len * n);
    let A
        = len
          / (2 * Math.tan((180 / n)
                     * 3.14159 / 180));
    let area = (P * A) / 2;
 
    return area;
}
 
// Function to find the area of a triangle
function area_of_triangle_inscribed( n,  len)
{
 
    let area = area_of_regular_polygon(n, len);
 
    // area of one triangle
    // in an N-sided regular polygon
    let triangle = area / n;
 
    // area of inscribed triangle
    let ins_tri = (triangle * 3);
 
    return ins_tri;
}
 
// Driver code
let  n = 6, len = 10;
 
  document.write( area_of_triangle_inscribed(n, len).toFixed(3));
 
 
// This code is contributed by todaysgaurav
 
</script>
Output: 
129.904

 

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