Shortest distance between a point and a circle

Last Updated : 08 May, 2019

Given a circle with a given radius has its centre at a particular position in the coordinate plane. In the coordinate plane, another point is given. The task is to find the shortest distance between the point and the circle.

Examples:

Input: x1 = 4, y1 = 6, x2 = 35, y2 = 42, r = 5 
Output: 42.5079

Input: x1 = 0, y1 = 0, x2 = 5, y2 = 12, r = 3
Output: 10

Approach:

Let the radius of the circle = r

co-ordinate of the centre of circle = (x1, y1)

co-ordinate of the point = (x2, y2)

let the distance between centre and the point = d

As the line AC intersects the circle at B, so the shortest distance will be BC,
which is equal to (d-r)

here using the distance formula,
d = √((x2-x1)^2 – (y2-y1)^2)

so BC = √((x2-x1)^2 – (y2-y1)^2) – r

so,
$Shortest distance between the point and the circle = sqrt((x2-x1)^2 - (y2-y1)^2) - r$

Below is the implementation of the above approach:

C++

// C++ program to find 
// the Shortest distance 
// between a point and 
// a circle 
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to find the shortest distance 
void dist(double x1, double y1, double x2, double y2, double r) 
{ 
    cout << "The shortest distance "
         << "between a point and a circle is "
         << sqrt((pow((x2 - x1), 2)) 
                 + (pow((y2 - y1), 2))) 
                - r 
         << endl; 
} 
  
// Driver code 
int main() 
{ 
    double x1 = 4, y1 = 6, 
           x2 = 35, y2 = 42, r = 5; 
    dist(x1, y1, x2, y2, r); 
    return 0; 
} 

Java

// Java program to find 
// the Shortest distance 
// between a point and 
// a circle 
class GFG 
{ 
  
// Function to find the shortest distance 
static void dist(double x1, double y1, double x2, 
                                double y2, double r) 
{ 
    System.out.println("The shortest distance "
            + "between a point and a circle is "
            + (Math.sqrt((Math.pow((x2 - x1), 2)) 
                    + (Math.pow((y2 - y1), 2))) 
            - r)); 
} 
  
// Driver code 
public static void main(String[] args) 
{ 
    double x1 = 4, y1 = 6, 
            x2 = 35, y2 = 42, r = 5; 
    dist(x1, y1, x2, y2, r); 
} 
} 
  
/* This code contributed by PrinciRaj1992 */

Python3

# Python program to find  
# the Shortest distance  
# between a point and  
# a circle  
   
# Function to find the shortest distance  
def dist(x1, y1, x2, y2, r):  
    print("The shortest distance between a point and a circle is "
    ,((((x2 - x1)** 2) + ((y2 - y1)** 2))**(1/2)) - r); 
  
   
# Driver code  
x1 = 4; 
y1 = 6;  
x2 = 35; 
y2 = 42; 
r = 5;  
dist(x1, y1, x2, y2, r);  
  
  
# This code has been contributed by 29AjayKumar 

C#

// C# program to find the Shortest distance 
// between a point and a circle 
using System; 
  
class GFG 
{ 
  
// Function to find the shortest distance 
static void dist(double x1, double y1, double x2, 
                                double y2, double r) 
{ 
    Console.WriteLine("The shortest distance "
            + "between a point and a circle is "
            + (Math.Sqrt((Math.Pow((x2 - x1), 2)) 
                    + (Math.Pow((y2 - y1), 2))) 
            - r)); 
} 
  
// Driver code 
public static void Main(String[] args) 
{ 
    double x1 = 4, y1 = 6, 
            x2 = 35, y2 = 42, r = 5; 
    dist(x1, y1, x2, y2, r); 
} 
} 
  
/* This code contributed by PrinciRaj1992 */

PHP

<?php 
// PHP program to find  
// the Shortest distance  
// between a point and  
// a circle  
  
// Function to find the shortest distance  
function dist($x1, $y1, $x2, $y2, $r)  
{  
    echo "The shortest distance between a point and a circle is "
                ,sqrt((pow(($x2 - $x1), 2))  
                + (pow(($y2 - $y1), 2)))  
                - $r ; 
}  
  
// Driver code  
$x1 = 4; 
$y1 = 6;  
$x2 = 35; 
$y2 = 42; 
$r = 5;  
dist($x1, $y1, $x2, $y2, $r);  
  
// This code is contributed by AnkitRai01 
  
?> 

Output:

The shortest distance between a point and a circle is 42.5079

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