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++
// 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 polygondouble 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 triangledouble 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 codeint 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 polygonimport java.util.*;class GFG{// Function to find the area of the polygonstatic 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 trianglestatic 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 codestatic 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 = 6len = 10print(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 polygonusing System; class GFG{// Function to find the area of the polygonstatic 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 trianglestatic 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 codestatic 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 polygonfunction 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 trianglefunction 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 codelet n = 6, len = 10; document.write( area_of_triangle_inscribed(n, len).toFixed(3));// This code is contributed by todaysgaurav </script> |
Output:
129.904
Time Complexity: O(1), the code will run in a constant time.
Auxiliary Space: O(1), no extra space is required, so it is a constant.
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