# Circle and Lattice Points

Given a circle of radius **r** in 2-D with origin or (0, 0) as center. The task is to find the total lattice points on circumference. Lattice Points are points with coordinates as integers in 2-D space.

Example:

Input : r = 5. Output : 12 Below are lattice points on a circle with radius 5 and origin as (0, 0). (0,5), (0,-5), (5,0), (-5,0), (3,4), (-3,4), (-3,-4), (3,-4), (4,3), (-4,3), (-4,-3), (4,-3). are 12 lattice point.

To find lattice points, we basically need to find values of (x, y) which satisfy the equation x^{2} + y^{2} = r^{2}.

For any value of (x, y) that satisfies the above equation we actually have total 4 different combination which that satisfy the equation. For example if r = 5 and (3, 4) is a pair which satisfies the equation, there are actually 4 combinations (3, 4) , (-3,4) , (-3,-4) , (3,-4). There is an exception though, in case of (0, r) or (r, 0) there are actually 2 points as there is no negative 0.

// Initialize result as 4 for (r, 0), (-r. 0), // (0, r) and (0, -r) result = 4 Loop for x = 1 to r-1 and do following for every x. If r*r - x*x is a perfect square, then add 4 tor result.

Below is the implementation of above idea.

## CPP

`// C++ program to find countLattice points on a circle` `#include<bits/stdc++.h>` `using` `namespace` `std;` `// Function to count Lattice points on a circle` `int` `countLattice(` `int` `r)` `{` ` ` `if` `(r <= 0)` ` ` `return` `0; ` ` ` `// Initialize result as 4 for (r, 0), (-r. 0),` ` ` `// (0, r) and (0, -r)` ` ` `int` `result = 4;` ` ` `// Check every value that can be potential x` ` ` `for` `(` `int` `x=1; x<r; x++)` ` ` `{` ` ` `// Find a potential y` ` ` `int` `ySquare = r*r - x*x;` ` ` `int` `y = ` `sqrt` `(ySquare);` ` ` `// checking whether square root is an integer` ` ` `// or not. Count increments by 4 for four ` ` ` `// different quadrant values` ` ` `if` `(y*y == ySquare)` ` ` `result += 4;` ` ` `}` ` ` `return` `result;` `}` `// Driver program` `int` `main()` `{` ` ` `int` `r = 5;` ` ` `cout << countLattice(r);` ` ` `return` `0;` `}` |

## Java

`// Java program to find` `// countLattice points on a circle` `class` `GFG` `{` `// Function to count` `// Lattice points on a circle` `static` `int` `countLattice(` `int` `r)` `{` ` ` `if` `(r <= ` `0` `)` ` ` `return` `0` `; ` ` ` ` ` `// Initialize result as 4 for (r, 0), (-r. 0),` ` ` `// (0, r) and (0, -r)` ` ` `int` `result = ` `4` `;` ` ` ` ` `// Check every value that can be potential x` ` ` `for` `(` `int` `x=` `1` `; x<r; x++)` ` ` `{` ` ` `// Find a potential y` ` ` `int` `ySquare = r*r - x*x;` ` ` `int` `y = (` `int` `)Math.sqrt(ySquare);` ` ` ` ` `// checking whether square root is an integer` ` ` `// or not. Count increments by 4 for four ` ` ` `// different quadrant values` ` ` `if` `(y*y == ySquare)` ` ` `result += ` `4` `;` ` ` `}` ` ` ` ` `return` `result;` `}` `// Driver code` `public` `static` `void` `main(String arg[]) ` `{` ` ` `int` `r = ` `5` `;` ` ` `System.out.println(countLattice(r));` `}` `}` `// This code is contributed by Anant Agarwal.` |

## Python3

`# Python3 program to find` `# countLattice points on a circle` `import` `math` `# Function to count Lattice` `# points on a circle` `def` `countLattice(r):` ` ` `if` `(r <` `=` `0` `):` ` ` `return` `0` ` ` `# Initialize result as 4 for (r, 0), (-r. 0),` ` ` `# (0, r) and (0, -r)` ` ` `result ` `=` `4` ` ` `# Check every value that can be potential x` ` ` `for` `x ` `in` `range` `(` `1` `, r):` ` ` ` ` `# Find a potential y` ` ` `ySquare ` `=` `r` `*` `r ` `-` `x` `*` `x ` ` ` `y ` `=` `int` `(math.sqrt(ySquare)) ` ` ` `# checking whether square root is an integer` ` ` `# or not. Count increments by 4 for four ` ` ` `# different quadrant values` ` ` `if` `(y` `*` `y ` `=` `=` `ySquare):` ` ` `result ` `+` `=` `4` ` ` ` ` `return` `result ` ` ` `# Driver program` `r ` `=` `5` `print` `(countLattice(r)) ` `# This code is contributed by` `# Smitha Dinesh Semwal` |

## C#

`// C# program to find countLattice` `// points on a circle` `using` `System;` `class` `GFG {` ` ` `// Function to count Lattice` ` ` `// points on a circle` ` ` `static` `int` `countLattice(` `int` `r)` ` ` `{` ` ` `if` `(r <= 0)` ` ` `return` `0; ` ` ` ` ` `// Initialize result as 4` ` ` `// for (r, 0), (-r. 0),` ` ` `// (0, r) and (0, -r)` ` ` `int` `result = 4;` ` ` ` ` `// Check every value that` ` ` `// can be potential x` ` ` `for` `(` `int` `x = 1; x < r; x++)` ` ` `{` ` ` ` ` `// Find a potential y` ` ` `int` `ySquare = r*r - x*x;` ` ` `int` `y = (` `int` `)Math.Sqrt(ySquare);` ` ` ` ` `// checking whether square root` ` ` `// is an integer or not. Count` ` ` `// increments by 4 for four ` ` ` `// different quadrant values` ` ` `if` `(y*y == ySquare)` ` ` `result += 4;` ` ` `}` ` ` ` ` `return` `result;` ` ` `}` ` ` ` ` `// Driver code` ` ` `public` `static` `void` `Main() ` ` ` `{` ` ` `int` `r = 5;` ` ` ` ` `Console.Write(countLattice(r));` ` ` `}` `}` `// This code is contributed by nitin mittal.` |

## PHP

`<?php` `// PHP program to find countLattice` `// points on a circle` `// Function to count Lattice ` `// points on a circle` `function` `countLattice(` `$r` `)` `{` ` ` `if` `(` `$r` `<= 0)` ` ` `return` `0; ` ` ` `// Initialize result as 4 ` ` ` `// for (r, 0), (-r. 0),` ` ` `// (0, r) and (0, -r)` ` ` `$result` `= 4;` ` ` `// Check every value that ` ` ` `// can be potential x` ` ` `for` `(` `$x` `= 1; ` `$x` `< ` `$r` `; ` `$x` `++)` ` ` `{` ` ` ` ` `// Find a potential y` ` ` `$ySquare` `= ` `$r` `* ` `$r` `- ` `$x` `* ` `$x` `;` ` ` `$y` `= ` `ceil` `(sqrt(` `$ySquare` `));` ` ` `// checking whether square ` ` ` `// root is an integer` ` ` `// or not. Count increments` ` ` `// by 4 for four different` ` ` `// quadrant values` ` ` `if` `(` `$y` `* ` `$y` `== ` `$ySquare` `)` ` ` `$result` `+= 4;` ` ` `}` ` ` `return` `$result` `;` `}` ` ` `// Driver Code` ` ` `$r` `= 5;` ` ` `echo` `countLattice(` `$r` `);` `// This code is contributed by nitin mittal` `?>` |

## Javascript

`<script>` `// javascript program to find` `// countLattice points on a circle` ` ` `// Function to count` ` ` `// Lattice points on a circle` ` ` `function` `countLattice(r) {` ` ` `if` `(r <= 0)` ` ` `return` `0;` ` ` `// Initialize result as 4 for (r, 0), (-r. 0),` ` ` `// (0, r) and (0, -r)` ` ` `var` `result = 4;` ` ` `// Check every value that can be potential x` ` ` `for` `(x = 1; x < r; x++)` ` ` `{` ` ` ` ` `// Find a potential y` ` ` `var` `ySquare = r * r - x * x;` ` ` `var` `y = parseInt( Math.sqrt(ySquare));` ` ` `// checking whether square root is an integer` ` ` `// or not. Count increments by 4 for four` ` ` `// different quadrant values` ` ` `if` `(y * y == ySquare)` ` ` `result += 4;` ` ` `}` ` ` `return` `result;` ` ` `}` ` ` `// Driver code` ` ` `var` `r = 5;` ` ` `document.write(countLattice(r));` `// This code is contributed by umadevi9616 ` `</script>` |

Output:

12

**Time Complexity:** O(1)**Auxiliary Space:** O(1)

**Reference:**

http://mathworld.wolfram.com/CircleLatticePoints.html

This article is contributed by **Shivam Pradhan (anuj_charm)**. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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