# Distance between a point and a Plane in 3 D

You are given a points (x1, y1, z1) and a plane a * x + b * y + c * z + d = 0. The task is to find the perpendicular(shortest) distance between that point and the given Plane.

**Examples :**

Input:x1 = 4, y1 = -4, z1 = 3, a = 2, b = -2, c = 5, d = 8Output:Perpendicular distance is 6.78902858227Input:x1 = 2, y1 = 8, z1 = 5, a = 1, b = -2, c = -2, d = -1Output:Perpendicular distance is 8.33333333333

**Approach: **The perpendicular distance (i.e shortest distance) from a given point to a Plane is the perpendicular distance from that point to the given plane. Let the co-ordinate of the given point be (x1, y1, z1)

and equation of the plane be given by the equation a * x + b * y + c * z + d = 0, where a, b and c are real constants.

The formula for distance between a point and Plane in 3-D is given by:

Distance = (| a*x1 + b*y1 + c*z1 + d |) / (sqrt( a*a + b*b + c*c))

Below is the implementation of the above formulae:

## C++

`// C++ program to find the` `// Perpendicular(shortest)` `// distance between a point` `// and a Plane in 3 D.` `#include<bits/stdc++.h>` `#include<math.h>` `using` `namespace` `std;` `// Function to find distance` `void` `shortest_distance(` `float` `x1, ` `float` `y1,` ` ` `float` `z1, ` `float` `a,` ` ` `float` `b, ` `float` `c,` ` ` `float` `d)` `{` ` ` `d = ` `fabs` `((a * x1 + b * y1 +` ` ` `c * z1 + d));` ` ` `float` `e = ` `sqrt` `(a * a + b *` ` ` `b + c * c);` ` ` `cout << ` `"Perpendicular distance is "` ` ` `<< (d / e);` ` ` `return` `;` `}` `// Driver Code` `int` `main()` `{` ` ` `float` `x1 = 4;` ` ` `float` `y1 = -4;` ` ` `float` `z1 = 3;` ` ` `float` `a = 2;` ` ` `float` `b = -2;` ` ` `float` `c = 5;` ` ` `float` `d = 8;` ` ` `// Function call` ` ` `shortest_distance(x1, y1, z1,` ` ` `a, b, c, d);` `}` `// This code is contributed` `// by Akanksha Rai(Abby_akku)` |

## C

`// C program to find the Perpendicular(shortest)` `// distance between a point and a Plane in 3 D.` `#include<stdio.h>` `#include<math.h>` `// Function to find distance` `void` `shortest_distance(` `float` `x1, ` `float` `y1, ` `float` `z1,` ` ` `float` `a, ` `float` `b, ` `float` `c, ` `float` `d)` `{` ` ` `d = ` `fabs` `((a * x1 + b * y1 + c * z1 + d));` ` ` `float` `e = ` `sqrt` `(a * a + b * b + c * c);` ` ` `printf` `(` `"Perpendicular distance is %f"` `, d/e);` ` ` `return` `;` `}` `// Driver Code` `int` `main()` `{` ` ` `float` `x1 = 4;` ` ` `float` `y1 = -4;` ` ` `float` `z1 = 3;` ` ` `float` `a = 2;` ` ` `float` `b = -2;` ` ` `float` `c = 5;` ` ` `float` `d = 8;` ` ` `// Function call` ` ` `shortest_distance(x1, y1, z1, a, b, c, d);` `}` `// This code is contributed` `// by Amber_Saxena.` |

## Java

`// Java program to find the` `// Perpendicular(shortest)` `// distance between a point` `// and a Plane in 3 D.` `import` `java .io.*;` `class` `GFG` `{` ` ` `// Function to find distance` `static` `void` `shortest_distance(` `float` `x1, ` `float` `y1,` ` ` `float` `z1, ` `float` `a,` ` ` `float` `b, ` `float` `c,` ` ` `float` `d)` `{` ` ` `d = Math.abs((a * x1 + b *` ` ` `y1 + c * z1 + d));` ` ` `float` `e = (` `float` `)Math.sqrt(a * a + b *` ` ` `b + c * c);` ` ` `System.out.println(` `"Perpendicular distance "` `+` ` ` `"is "` `+ d / e);` `}` `// Driver code` `public` `static` `void` `main(String[] args)` `{` ` ` `float` `x1 = ` `4` `;` ` ` `float` `y1 = -` `4` `;` ` ` `float` `z1 = ` `3` `;` ` ` `float` `a = ` `2` `;` ` ` `float` `b = -` `2` `;` ` ` `float` `c = ` `5` `;` ` ` `float` `d = ` `8` `;` ` ` `// Function call` ` ` `shortest_distance(x1, y1, z1,` ` ` `a, b, c, d);` `}` `}` `// This code is contributed` `// by Amber_Saxena.` |

## Python

`# Python program to find the Perpendicular(shortest)` `# distance between a point and a Plane in 3 D.` `import` `math` `# Function to find distance` `def` `shortest_distance(x1, y1, z1, a, b, c, d):` ` ` ` ` `d ` `=` `abs` `((a ` `*` `x1 ` `+` `b ` `*` `y1 ` `+` `c ` `*` `z1 ` `+` `d))` ` ` `e ` `=` `(math.sqrt(a ` `*` `a ` `+` `b ` `*` `b ` `+` `c ` `*` `c))` ` ` `print` `(` `"Perpendicular distance is"` `, d` `/` `e)` ` ` `# Driver Code` `x1 ` `=` `4` `y1 ` `=` `-` `4` `z1 ` `=` `3` `a ` `=` `2` `b ` `=` `-` `2` `c ` `=` `5` `d ` `=` `8` `# Function call` `shortest_distance(x1, y1, z1, a, b, c, d) ` |

## C#

`// C# program to find the` `// Perpendicular(shortest)` `// distance between a point` `// and a Plane in 3 D.` `using` `System;` `class` `GFG` `{` ` ` `// Function to find distance` `static` `void` `shortest_distance(` `float` `x1, ` `float` `y1,` ` ` `float` `z1, ` `float` `a,` ` ` `float` `b, ` `float` `c,` ` ` `float` `d)` `{` ` ` `d = Math.Abs((a * x1 + b *` ` ` `y1 + c * z1 + d));` ` ` `float` `e = (` `float` `)Math.Sqrt(a * a + b *` ` ` `b + c * c);` ` ` `Console.Write(` `"Perpendicular distance "` `+` ` ` `"is "` `+ d / e);` `}` `// Driver code` `public` `static` `void` `Main()` `{` ` ` `float` `x1 = 4;` ` ` `float` `y1 = -4;` ` ` `float` `z1 = 3;` ` ` `float` `a = 2;` ` ` `float` `b = -2;` ` ` `float` `c = 5;` ` ` `float` `d = 8;` ` ` `// Function call` ` ` `shortest_distance(x1, y1, z1,` ` ` `a, b, c, d);` `}` `}` `// This code is contributed` `// by ChitraNayal` |

## PHP

`<?php` `// PHP program to find the` `// Perpendicular(shortest)` `// distance between a point` `// and a Plane in 3 D.` `// Function to find distance` `function` `shortest_distance(` `$x1` `, ` `$y1` `, ` `$z1` `,` ` ` `$a` `, ` `$b` `, ` `$c` `, ` `$d` `)` `{` ` ` `$d` `= ` `abs` `((` `$a` `* ` `$x1` `+ ` `$b` `* ` `$y1` `+` ` ` `$c` `* ` `$z1` `+ ` `$d` `));` ` ` `$e` `= sqrt(` `$a` `* ` `$a` `+ ` `$b` `*` ` ` `$b` `+ ` `$c` `* ` `$c` `);` ` ` `echo` `"Perpendicular distance is "` `. ` `$d` `/ ` `$e` `;` `}` `// Driver Code` `$x1` `= 4;` `$y1` `= -4;` `$z1` `= 3;` `$a` `= 2;` `$b` `= -2;` `$c` `= 5;` `$d` `= 8;` ` ` `// function call` `shortest_distance(` `$x1` `, ` `$y1` `, ` `$z1` `,` ` ` `$a` `, ` `$b` `, ` `$c` `, ` `$d` `);` `// This code is contributed` `// by Amber_Saxena.` `?>` |

## Javascript

`<script>` `// Javascript program to find the` `// Perpendicular(shortest)` `// distance between a point` `// and a Plane in 3 D.` `// Function to find distance` `function` `shortest_distance( x1, y1, z1, a,` ` ` `b, c, d)` `{` ` ` `d = Math.abs((a * x1 + b * y1 +` ` ` `c * z1 + d));` ` ` `let e = Math.sqrt(a * a + b *` ` ` `b + c * c);` ` ` `document.write(` `"Perpendicular distance is "` ` ` `+ (d / e));` ` ` `return` `;` `}` ` ` `// driver code ` ` ` `let x1 = 4;` ` ` `let y1 = -4;` ` ` `let z1 = 3;` ` ` `let a = 2;` ` ` `let b = -2;` ` ` `let c = 5;` ` ` `let d = 8;` ` ` `// Function call` ` ` `shortest_distance(x1, y1, z1,` ` ` `a, b, c, d);` ` ` ` ` `// This code is contributed by jana_sayantan. ` `</script>` |

**Output:**

Perpendicular distance is 6.78902858227