寻找圆段面积的程序

在一个圆中，如果绘制了一个和弦，那么该和弦会将整个圆分成两部分。圆的这两部分称为圆的段。较小的区域称为小段，较大的区域称为大段。
在下图中，和弦 AB 将圆分成小段和大段。

我们给定了圆的半径和形成小段的角度。我们需要找到两个部分的区域。
例子 ：

Input : 
radius = 21.0
angle = 120.0
Output :
Area of minor segment 270.855
Area of major segment 1114.59

Input :
radius = 10.0
angle = 90.0
Output : 
Area of minor segment 28.5397
Area of major segment 285.619

段面积：

为此，我们将弦的端点与圆心连接起来，形成一个扇形，该扇形在中心处对着某个“角度”。并且从弦AB上的圆心绘制垂线。通过三角形的全等，我们得到 ∠ AOP = ∠ BOP = 1/2（角度）。

线段面积公式：

Area of Segment = Area of sector - Area of Triangle OAB 
                = pi * r2 * (angle/360) -
                  Area of Triangle OAB

有关扇区面积公式的详细信息，请参阅 https://www.geeksforgeeks.org/area-of-a-sector/。

In the figure above, assume angle made by sector = X,
so ∠ AOP = ∠ BOP = X/2

Area of Triangle AOB = 1/2 * base * height
                     = 1/2 * AB * OP

Now in Triangle AOP, By trigonometry
Cos(X/2) = OP/AO  i.e. OP = AO * Cos(X/2) 
                       OP = r * Cos(X/2)
Sin(X/2) = AP/AO  i.e. AP = AO * Sin(X/2) 
                       AP = r * Sin(X/2)

So,
Base = AB = AP + PB
          = 2 * AP
          = 2 * r * Sin(X/2)
   
Height = OP = r * Cos(X/2)

Area of triangle = 1/2 * (2 * r * Sin(X/2)) * (r * Cos(X/2))
                    = 1/2 * r2 * Sin(X) 
                      [Using identity 2 * Sin(A) * Cos(A)]
                    = Sin(2 * A))

Hence Area of Segment =  pi * r2 * (angle/360) - 1/2 * r2 * Sin(angle)

C++

// C++ Program to
// find area of
// segment of a
// circle
#include 
using namespace std;
 
float pi = 3.14159;
 
// Function to find
// area of segment
float area_of_segment(float radius,
                      float angle)
{
    // Calculating area of sector
    float area_of_sector = pi *
                           (radius * radius)
                           *(angle / 360);
 
    // Calculating area of triangle
    float area_of_triangle = (float)1 / 2 *
                             (radius * radius) *
                             sin((angle * pi) / 180);
 
    return area_of_sector - area_of_triangle;
}
 
// Driver Code
int main()
{
    float radius = 10.0, angle = 90.0;
    cout << "Area of minor segment = "
        << area_of_segment(radius, angle) << endl;
 
    cout << "Area of major segment = "
        << area_of_segment(radius, (360 - angle));
}

Java

// Java Program to find area of
// segment of a circle
class GFG {
static float pi = 3.14159f;
 
static float area_of_segment(float radius,
                              float angle)
{
    // Calculating area of sector
    float area_of_sector = pi *
   (radius * radius) * (angle / 360);
 
    // Calculating area of triangle
    float area_of_triangle =
        (float)1 / 2 * (radius * radius) *
        (float)Math.sin((angle * pi) / 180);
 
    return area_of_sector - area_of_triangle;
}
 
// Driver Function
public static void main(String[] args)
{
    float radius = 10.0f, angle = 90.0f;
    System.out.println("Area of minor segment = " +
                    area_of_segment(radius, angle));
 
    System.out.println("Area of major segment = " +
                    area_of_segment(radius, (360 - angle)));
}
}
 
// This code is contributed by Anant Agarwal.

Python

# Python3 Program
# to find area of
# segment of a
# circle
import math
 
pi = 3.14159
 
# Function to find
# area of segment
def area_of_segment(radius, angle):
    # Calculating area of sector
    area_of_sector = pi *
                     (radius * radius)
                     * (angle / 360)
 
    # Calculating area of triangle
    area_of_triangle = 1 / 2 *
                       (radius * radius) *
                       math.sin((angle * pi) / 180)
 
    return area_of_sector - area_of_triangle;
 
 
# Driver Code
radius = 10.0
angle = 90.0
print("Area of minor segment =",
       area_of_segment(radius, angle))
print("Area of major segment =",
      area_of_segment(radius, (360 - angle)))
       
      
# This code is contributed by Smitha Dinesh Semwal

C#

// C# Program to find area
// of segment of a circle
using System;
 
class GFG {
      static float pi = 3.14159f;
 
      static float area_of_segment(float radius,
                                   float angle)
{
    // Calculating area of sector
    float area_of_sector = pi * (radius * radius)
                             * (angle / 360);
 
    // Calculating area of triangle
    float area_of_triangle =(float)1 / 2 * (radius * radius)
                            *(float)Math.Sin((angle * pi) / 180);
 
    return area_of_sector - area_of_triangle;
}
 
    // Driver Function
    public static void Main()
{
    float radius = 10.0f, angle = 90.0f;
    Console.WriteLine("Area of minor segment = " +
                        area_of_segment(radius, angle));
 
    Console.WriteLine("Area of major segment = " +
                       area_of_segment(radius, (360 - angle)));
}
}
 
// This code is contributed by vt_m.

Javascript

 
// javascript program to find area of
// segment of a circle
let pi = 3.14159;
 
function area_of_segment(radius,
                              angle)
{
    // Calculating area of sector
    let area_of_sector = pi *
   (radius * radius) * (angle / 360);
   
    // Calculating area of triangle
    let area_of_triangle =
        1 / 2 * (radius * radius) *
        Math.sin((angle * pi) / 180);
   
    return area_of_sector - area_of_triangle;
}
   
 
// Driver Function
 
         let radius = 10.0, angle = 90.0;
    document.write("Area of minor segment = " +
                    area_of_segment(radius, angle) + "<br/>");
   
    document.write("Area of major segment = " +
                    area_of_segment(radius, (360 - angle)));
     
    // This code is contributed by jana_sayantan.

输出 ：

Area of minor segment = 28.5397
Area of major segment = 285.619