# 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.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.