在一个圆中,如果绘制了一个和弦,则该和弦会将整个圆分为两部分。圆的这两个部分称为圆的段。较小的区域称为次要部分,较大的区域称为主要部分。
在下图中,和弦AB将圆分为小段和大段。
给定形成较小段的圆弧半径和角度。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
细分区域:
为此,我们将和弦的端点与圆心连接在一起,形成一个扇形,该扇形在中心处有一些“角度”。并从和弦AB上的圆心绘制垂直线。通过三角形的全等,我们获得∠AOP =∠BOP = 1/2(角度)。
细分区域的公式:
Area of Segment = Area of sector - Area of Triangle OAB
= pi * r2 * (angle/360) -
Area of Triangle OAB
有关部门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.
PHP
Javascript
输出 :
Area of minor segment = 28.5397
Area of major segment = 285.619