内接于半径为 R 的圆的等边三角形的面积

给定一个表示圆半径的整数R ，任务是找到与该圆内接的等边三角形的面积。

例子：

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 

方法：让上面显示的三角形是一个等边三角形，表示为PQR 。

• 三角形的面积可以计算为：

Area of triangle = (1/2) * Base * Height

• 在这种情况下，Base 可以是PQ、PR 或 QR ，三角形的高度可以是PM 。因此，

Area of Triangle = (1/2) * QR * PM

• 现在在三角形 ORQ上应用正弦定律，

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

• 因此，所需等边三角形的底和高将是：

Base = r * sqrt(3) = r * 1.732
Height = (3/2) * r

• 借助上面给出的公式计算三角形的面积。

下面是上述方法的实现：

C++

// 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

// 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

 
// 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
 

输出：

63.651

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