Area of Equilateral triangle inscribed in a Circle of radius R

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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++

// 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

Java

// 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

Python3

# 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#

// 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

Javascript

<script>
 
// Javascript 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
function area(R) 
{
     
    // Base and Height of
    // equilateral triangle
    var base = 1.732 * R;
    var height = (1.5) * R;
 
    // Area using Base and Height
    var area = 0.5 * base * height;
    return area;
}
 
// Driver code
var R = 7;
 
document.write(area(R));
 
// This code is contributed by todaysgaurav 
 
</script>

Output:

63.651

Time complexity : O(1)
Auxiliary Space : O(1)

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