# Angle subtended by an arc at the centre of a circle

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)

- So
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; ` `} ` |

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## 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. ` |

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## 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 ` |

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## 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 ` |

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**Output:**

60

**Time Complexity:** O(1)

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