# Area of a Regular Pentagram

• Difficulty Level : Medium
• Last Updated : 04 Aug, 2021

Given a Pentagram and it’s inner side length(d). The task is find out area of Pentagram. The Pentagram is a five-pointed star that is formed by drawing a continuous line in five straight segments. Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.  To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

In case you wish to attend live classes with experts, please refer DSA Live Classes for Working Professionals and Competitive Programming Live for Students.

Examples:

Input: d = 5
Output: Area = 139.187
Area of regular pentagram = 139.187

Input: d = 7
Output: Area = 272.807

Idea is to use Golden Ratio between a/b, b/c, and c/d which equals approximately 1.618
Inner side length d is given so
c = 1.618 * d
b = 1.618 * c
a = 1.618 * b
AB, BC and CD are equals(both side of regular pentagram)
So AB = BC = CD = c and BD is given by d.

Area of pentagram = Area of Pentagon BDFHJ + 5 * (Area of triangle BCD)
Area of Pentagon BDFHJ = (d^2 * 5)/ (4* tan 36)
Area of triangle BCD = [s(s-d)(s-c)(s-c)]^(1/2) {Heron’s Formula}
where
s = (d + c + c)/2

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach``#include ``#define PI 3.14159``using` `namespace` `std;` `// Function to return the area of triangle BCD``double` `areaOfTriangle(``float` `d)``{``    ``// Using Golden ratio``    ``float` `c = 1.618 * d;``    ``float` `s = (d + c + c) / 2;` `    ``// Calculate area of triangle BCD``    ``double` `area = ``sqrt``(s * (s - c) *``                          ``(s - c) * (s - d));` `    ``// Return area of all 5 triangle are same``    ``return` `5 * area;``}` `// Function to return the area of regular pentagon``double` `areaOfRegPentagon(``float` `d)``{``    ``// Calculate the area of regular``    ``// pentagon using above formula``    ``double` `cal = 4 * ``tan``(PI / 5);``    ``double` `area = (5 * d * d) / cal;` `    ``// Return area of regular pentagon``    ``return` `area;``}` `// Function to return the area of pentagram``double` `areaOfPentagram(``float` `d)``{``    ``// Area of a pentagram is equal to the``    ``// area of regular  pentagon and five times``    ``// the area of Triangle``    ``return` `areaOfRegPentagon(d) +``                             ``areaOfTriangle(d);``}` `// Driver code``int` `main()``{``    ``float` `d = 5;``    ``cout << areaOfPentagram(d) << endl;` `    ``return` `0;``}`

## Java

 `// Java implementation of above approach``public` `class` `GFG``{` `    ``static` `double` `PI = ``3.14159``;` `    ``// Function to return the area of triangle BCD``    ``static` `double` `areaOfTriangle(``float` `d)``    ``{``        ``// Using Golden ratio``        ``float` `c = (``float``) (``1.618` `* d);``        ``float` `s = (d + c + c) / ``2``;` `        ``// Calculate area of triangle BCD``        ``double` `area = Math.sqrt(s * (s - c)``                ``* (s - c) * (s - d));` `        ``// Return area of all 5 triangle are same``        ``return` `5` `* area;``    ``}` `    ``// Function to return the area of regular pentagon``    ``static` `double` `areaOfRegPentagon(``float` `d)``    ``{``        ``// Calculate the area of regular``        ``// pentagon using above formula``        ``double` `cal = ``4` `* Math.tan(PI / ``5``);``        ``double` `area = (``5` `* d * d) / cal;` `        ``// Return area of regular pentagon``        ``return` `area;``    ``}` `    ``// Function to return the area of pentagram``    ``static` `double` `areaOfPentagram(``float` `d)``    ``{``        ``// Area of a pentagram is equal to the``        ``// area of regular pentagon and five times``        ``// the area of Triangle``        ``return` `areaOfRegPentagon(d)``                ``+ areaOfTriangle(d);``    ``}` `    ``// Driver code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``float` `d = ``5``;``        ``System.out.println(areaOfPentagram(d));``    ``}``}` `// This code has been contributed by 29AjayKumar`

## Python3

 `# Python3 implementation of the approach` `import` `math` `PI ``=` `3.14159` `# Function to return the area of triangle BCD``def` `areaOfTriangle(d) :` `    ``# Using Golden ratio``    ``c ``=` `1.618` `*` `d``    ``s ``=` `(d ``+` `c ``+` `c) ``/` `2` `    ``# Calculate area of triangle BCD``    ``area ``=` `math.sqrt(s ``*` `(s ``-` `c) ``*``                        ``(s ``-` `c) ``*` `(s ``-` `d))` `    ``# Return area of all 5 triangles are the same``    ``return` `5` `*` `area`  `# Function to return the area of regular pentagon``def` `areaOfRegPentagon(d) :``    ` `    ``global` `PI``    ``# Calculate the area of regular``    ``# pentagon using above formula``    ``cal ``=` `4` `*` `math.tan(PI ``/` `5``)``    ``area ``=` `(``5` `*` `d ``*` `d) ``/` `cal``    ` `    ``# Return area of regular pentagon``    ``return` `area`  `# Function to return the area of pentagram``def` `areaOfPentagram(d) :` `    ``# Area of a pentagram is equal to the``    ``# area of regular pentagon and five times``    ``# the area of Triangle``    ``return` `areaOfRegPentagon(d) ``+` `areaOfTriangle(d)`  `# Driver code` `d ``=` `5``print``(areaOfPentagram(d))` `    ` `# This code is contributed by ihritik`

## C#

 `// C# implementation of the above approach``using` `System;` `class` `GFG``{` `    ``static` `double` `PI = 3.14159;` `    ``// Function to return the area of triangle BCD``    ``static` `double` `areaOfTriangle(``float` `d)``    ``{``        ``// Using Golden ratio``        ``float` `c = (``float``) (1.618 * d);``        ``float` `s = (d + c + c) / 2;` `        ``// Calculate area of triangle BCD``        ``double` `area = Math.Sqrt(s * (s - c)``                ``* (s - c) * (s - d));` `        ``// Return area of all 5 triangle are same``        ``return` `5 * area;``    ``}` `    ``// Function to return the area of regular pentagon``    ``static` `double` `areaOfRegPentagon(``float` `d)``    ``{``        ``// Calculate the area of regular``        ``// pentagon using above formula``        ``double` `cal = 4 * Math.Tan(PI / 5);``        ``double` `area = (5 * d * d) / cal;` `        ``// Return area of regular pentagon``        ``return` `area;``    ``}` `    ``// Function to return the area of pentagram``    ``static` `double` `areaOfPentagram(``float` `d)``    ``{``        ``// Area of a pentagram is equal to the``        ``// area of regular pentagon and five times``        ``// the area of Triangle``        ``return` `areaOfRegPentagon(d)``                ``+ areaOfTriangle(d);``    ``}` `    ``// Driver code``    ``public` `static` `void` `Main()``    ``{``        ``float` `d = 5;``        ``Console.WriteLine(areaOfPentagram(d));``    ``}``}` `// This code has been contributed by ihritik`

## Javascript

 ``
Output:
`139.187`

Time Complexity : O(1)

My Personal Notes arrow_drop_up