# 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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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++ Program to find the area of a triangle ` `// inscribed in N-sided regular polygon ` ` `  `#include ` `#include ` `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 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 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# 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 `

Output:
```129.904
```

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Improved By : divyamohan123, princiraj1992

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