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

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++

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// C++ implementation to find
// the area of the equilateral triangle
// inscribed in a circle of radius R
#include <iostream>
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

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Java

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// 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

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Python

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# 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))

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C#

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// 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

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Output:

63.651

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