Distance between Two Points Calculator

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.