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Angle subtended by an arc at the centre of a circle
  • Last Updated : 31 May, 2021

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)
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++




// 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;
}

Java




// 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.

Python3




# 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

C#




// 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

Javascript




<script>
// JavaScript implementation of the approach
 
// Function to find Angle
// subtended by an arc
// at the centre of a circle
function angle(n)
{
    return 2 * n;
}
 
// Driver code
 
    let n = 30;
    document.write(angle(n));
 
// This code is contributed by Surbhi Tyagi.
 
</script>
Output: 
60

 

Time Complexity: O(1)
 

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