Area of Equilateral triangle inscribed in a Circle of radius R

Given an integer R which denotes the radius of a circle, the task is to find the area of an equilateral triangle inscribed in this circle.

Examples:

Input: R = 4
Output: 20.784
Explanation:
Area of equilateral triangle inscribed in a circle of radius R will be 20.784, whereas side of the triangle will be 6.928

Input: R = 7
Output: 63.651
Explanation:
Area of equilateral triangle inscribed in a circle of radius R will be 63.651, whereas side of the triangle will be 12.124

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

Approach: Let the above triangle shown be an equilateral triangle denoted as PQR.

• The area of the triangle can be calculated as:
```Area of triangle = (1/2) * Base * Height
```
• In this case, Base can be PQ, PR or QR and The height of the triangle can be PM. Hence,
```Area of Triangle = (1/2) * QR * PM
```
• Now Applying sine law on the triangle ORQ,
``` RQ         OR
------  = -------
sin 60    sin 30

=> RQ = OR * sin60 / sin30
=> Side of Triangle = OR * sqrt(3)

As it is clearly observed
PM = PO + OM = r + r * sin30 = (3/2) * r
```
• Therefore, the Base and height of the required equilateral triangle will be:
```Base = r * sqrt(3) = r * 1.732
Height = (3/2) * r
```
• Compute the area of the triangle with the help of the formulae given above.

Below is the implementation of the above approach:

 `// C++ implementation to find ` `// the area of the equilateral triangle ` `// inscribed in a circle of radius R ` `#include ` `using` `namespace` `std; ` ` `  `// Function to find the area of ` `// equilateral triangle inscribed ` `// in a circle of radius R ` `double` `area(``int` `R) { ` `      `  `     ``// Base and Height of ` `    ``// equilateral triangle ` `    ``double` `base = 1.732 * R;  ` `    ``double` `height = (1.5) * R; ` `      `  `            ``// Area using Base and Height ` `    ``double` `area = 0.5 * base * height; ` `    ``return` `area; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `R = 7; ` `    ``cout<<(area(R)); ` `    ``return` `0; ` `} ` ` `  `// This code is contributed by 29AjayKumar `

 `// Java implementation to find ` `// the area of the equilateral triangle ` `// inscribed in a circle of radius R ` `class` `GFG ` `{ ` `    ``// Function to find the area of ` `    ``// equilateral triangle inscribed ` `    ``// in a circle of radius R ` `    ``static` `double` `area(``int` `R) { ` `         `  `                ``// Base and Height of ` `        ``// equilateral triangle ` `        ``double` `base = ``1.732` `* R;  ` `        ``double` `height = (``1.5``) * R; ` `         `  `                ``// Area using Base and Height ` `        ``double` `area = ``0.5` `* base * height; ` `        ``return` `area; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String[] args) { ` `        ``int` `R = ``7``; ` `        ``System.out.println(area(R)); ` ` `  `    ``} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

 `# Python 3 implementation to find ` `# the area of the equilateral triangle ` `# inscribed in a circle of radius R ` ` `  `# Function to find the area of  ` `# equilateral triangle inscribed ` `# in a circle of radius R ` `def` `area(R): ` `    ``# Base and Height of  ` `    ``# equilateral triangle ` `    ``base ``=` `1.732` `*` `R ` `    ``height ``=` `( ``3` `/` `2` `) ``*` `R ` `     `  `    ``# Area using Base and Height ` `    ``area ``=` `(( ``1` `/` `2` `) ``*` `base ``*` `height ) ` `    ``return` `area ` `     `  `# Driver Code ` `if` `__name__``=``=``'__main__'``: ` `    ``R ``=` `7` `    ``print``(area(R)) `

 `// C# implementation to find ` `// the area of the equilateral triangle ` `// inscribed in a circle of radius R ` `using` `System; ` ` `  `class` `GFG ` `{ ` `    ``// Function to find the area of ` `    ``// equilateral triangle inscribed ` `    ``// in a circle of radius R ` `    ``static` `double` `area(``int` `R)  ` `    ``{ ` `         `  `        ``// Base and Height of ` `        ``// equilateral triangle ` `        ``double` `Base = 1.732 * R;  ` `        ``double` `height = (1.5) * R; ` `         `  `        ``// Area using Base and Height ` `        ``double` `area = 0.5 * Base * height; ` `        ``return` `area; ` `    ``} ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main(String[] args) ` `    ``{ ` `        ``int` `R = 7; ` `        ``Console.WriteLine(area(R)); ` `    ``} ` `} ` ` `  `// This code is contributed by 29AjayKumar `

Output:
```63.651
```

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.

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Improved By : 29AjayKumar

Article Tags :
Practice Tags :