# Distance between Two Points Calculator

Enter the points in the required input fields and click calculate button to find the distance between two points

Formula:
|AB|= √ ( x2 - x1 ) + ( y2 - y1 )

X

Y

Point A:
Point B:

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## Distance between two points calculator

Know the length of the space between two points in a 2 dimensional plane using the distance between two points calculator. You will get the complete step-by-step distance calculation.

## How to Use This Distance Formula Calculator?

The directions to use this tool are given below:

1. Enter the X and Y coordinates of the point A.
2. Enter the X and Y coordinates of the point B.
3. Click Calculate.

## What is the distance formula?

In a Cartesian plane, the distance d between two points P1 (x1, y1) and P2 (x2, y2) is given by:

d = √[(x2 - x1)2 + (y2 - y1)2]

For a three-dimensional space, given two points  P1 (x1 ,y1, z1) and P2(x2, y2, z2), the distance d between these points is:

d = √ (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2

## Derivation:

The derivation of the distance formula can be explained using the Pythagorean theorem.

For the two-dimensional case:

Consider two points  P1 (x1, y1) and P2 (x2, y2) on a Cartesian plane.

Create a right triangle using the horizontal and vertical distances between these two points. The horizontal distance (the length of the base of the triangle) is |x2 -x1|  and the vertical distance (height of the triangle) is |y2 -y1|.

Using the Pythagorean theorem for a right triangle with sides a and b and hypotenuse c:

c2 = a2 + b2

In our case, a corresponds to the horizontal distance x2 - x1 and b corresponds to the vertical distance y2 - y1. Plugging in these values:

c2 = (x2 - x1)2 + (y2 - y1)2

Taking the square root of both sides gives us the distance d as:

d = √[(x2 - x1)2 + (y2 - y1)2]

For the three-dimensional case:

The process is analogous. Consider the distance between two points in each of the three dimensions (x, y, z) and use the Pythagorean theorem iteratively to derive the formula.

## How to find the distance between two points?

### Example:

A plane has two Points P1 (1,2) and P2 (4,6). Find the distance d between P1 and P2.

Solution:

Step 1: Find the difference between the x-coordinates.

Δx = x2 - x1

Δx = 4 − 1

Δx = 3

Step 2: Square the difference obtained in Step 1.

(Δx)2 = 32

(Δx)2 = 9

Step 3: Find the difference between the y-coordinates.

Δy = y2 - y1

Δy = 6 − 2

Δy = 4

Step 4: Square the difference obtained in Step 3.

(Δy)2 = 42

(Δy)2 = 16

Step 5: Add the squares of the differences from Steps 2 and 4.

Sum = (Δx)2 + (Δy)2

Sum = 9 + 16

Sum = 25

Step 6: Take the square root of the sum obtained in Step 5 to get the distance.

d = 25

d = 5

## Daily applications:

The distance formula, derived from the Pythagorean theorem, has a multitude of real-life applications. Here are some examples:

Navigation and Mapping: Before the era of GPS, sailors, and aviators would use basic principles of trigonometry and the distance formula to chart a course or determine their current location.

Sports: In sports like baseball or golf, one might use the distance formula to determine the distance a ball traveled from its starting point (where it was hit) to its ending point (where it landed).

Architecture and Construction: Architects and construction engineers use the distance formula to determine the distance between two points in a construction blueprint or model.