Angle subtended by an arc at the centre of a circle - GeeksforGeeks

10 Mar, 2022
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: 60Input: 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++
Java
Python3
C#
Javascript
// 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 circleint angle(int n){    return 2 * n;} // Driver codeint main(){    int n = 30;    cout << angle(n);     return 0;}
// Java implementation of the approachimport java.io.*; class GFG{     // Function to find Angle subtended// by an arc at the centre of a circlestatic int angle(int n){    return 2 * n;} // Driver codepublic static void main (String[] args){    int n = 30;    System.out.println(angle(n));}} // This code is contributed by ajit.
# Python3 implementation of the approach # Function to find Angle# subtended by an arc# at the centre of a circledef angle(n):    return 2 * n # Driver coden = 30print(angle(n)) # This code is contributed by Mohit Kumar
// C# implementation of the approachusing System; class GFG{     // Function to find Angle subtended// by an arc at the centre of a circlestatic int angle(int n){    return 2 * n;} // Driver codepublic static void Main(){    int n = 30;    Console.Write(angle(n));}} // This code is contributed by Akanksha_Rai
<script>// JavaScript implementation of the approach // Function to find Angle// subtended by an arc// at the centre of a circlefunction angle(n){    return 2 * n;} // Driver code     let n = 30;    document.write(angle(n)); // This code is contributed by Surbhi Tyagi. </script>
60
 
Time Complexity: O(1)
Auxiliary Space: O(1) 
mohit kumar 29
Akanksha_Rai
jit_t
surbhityagi15
bunnyram19
subham348
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