Angle subtended by an arc at the centre of a circle (original) (raw)

Last Updated : 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: 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

`

Time Complexity: O(1)

Auxiliary Space: O(1)