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> |
60
Time Complexity: O(1)
Auxiliary Space: O(1)
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