Angle subtended by an arc at the centre of a circle

Given the angle subtended by an arc at the circle circumference X, the task is to find the angle subtended by an arc at the centre of a circle.

For eg in the below given image, you are given angle X and you have to find angle Y.

Examples:



Input: X = 30
Output: 60

Input: X = 90
Output: 180

Approach:

  • When we draw the radius AD and the chord CB, we get three small triangles.
  • The three triangles ABC, ADB and ACD are isosceles as AB, AC and AD are radiuses of the circle.
  • So in each of these triangles, the two acute angles (s, t and u) in each are equal.
  • From the diagram, we can see
    D = t + u (i)
  • In triangle ABC,
    s + s + A = 180 (angles in triangle)
    ie, A = 180 - 2s  (ii)
    
  • In triangle BCD,
    (t + s) + (s + u) + (u + t) = 180 (angles in triangle again)
    so 2s + 2t + 2u = 180
    ie 2t + 2u = 180 - 2s (iii)
  • So
    A = 2t + 2u = 2D from (i), (ii)  and (iii)
  • Hence Proved that ‘the angle at the centre is twice the angle at the circumference‘.

Below is the implementation of the above approach:

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ implementation of the approach
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to find Angle
// subtended by an arc
// at the centre of a circle
int angle(int n)
{
    return 2 * n;
}
  
// Driver code
int main()
{
    int n = 30;
    cout << angle(n);
  
    return 0;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java implementation of the approach
import java.io.*;
  
class GFG
{
      
// Function to find Angle subtended 
// by an arc at the centre of a circle
static int angle(int n)
{
    return 2 * n;
}
  
// Driver code
public static void main (String[] args)
{
    int n = 30;
    System.out.println(angle(n));
}
}
  
// This code is contributed by ajit.

chevron_right


Python3

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python3 implementation of the approach
  
# Function to find Angle
# subtended by an arc
# at the centre of a circle
def angle(n):
    return 2 * n
  
# Driver code
n = 30
print(angle(n))
  
# This code is contributed by Mohit Kumar

chevron_right


C#

filter_none

edit
close

play_arrow

link
brightness_4
code

// C# implementation of the approach
using System;
  
class GFG
{
      
// Function to find Angle subtended 
// by an arc at the centre of a circle
static int angle(int n)
{
    return 2 * n;
}
  
// Driver code
public static void Main()
{
    int n = 30;
    Console.Write(angle(n));
}
}
  
// This code is contributed by Akanksha_Rai 

chevron_right


Output:

60

Time Complexity: O(1)



My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.