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 circleint angle(int n){ return 2 * n;}// Driver codeint main(){ int n = 30; cout << angle(n); return 0;} |
Java
// 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
# 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#
// 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 |
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)
Auxiliary Space: O(1)
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