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# Distance between Two Points Calculator

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

**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:

- Enter the
**X**and**Y**coordinates of the point**A**. - Enter the
**X**and**Y**coordinates of the point**B**. - Click
**Calculate**.

**What is the distance formula?**

In a Cartesian plane, the distance d between two points **P****1 (x _{1}, y_{1}**

**)**and

**P**

**2 (x**

_{2}, y_{2}**)**is given by:

**d = √[(x _{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}]**

For a three-dimensional space, given two points **P****1 (x _{1} ,y_{1}, z_{1}**

**)**and

**P**

**2(x**

_{2}, y_{2}, z_{2}**)**, the distance d between these points is:

**d = √ (x _{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}**

**Derivation:**

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

**For the two-dimensional case:**

Consider two points **P****1 (x _{1}, y_{1}**

**)**and

**P**

**2 (x**

_{2}, y_{2}**)**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 **|x _{2}**

**-x**

_{1}**|**and the

**vertical distance**(height of the triangle) is

**|y**

_{2}-y_{1}**|**.

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

**c ^{2} = a^{2 }+ b^{2}**

In our case, a corresponds to the **horizontal **distance **x**** _{2} - x_{1}** and

**b**corresponds to the vertical distance

**y**

_{2}**- y**. Plugging in these values:

_{1}**c ^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}**

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

**d = ****√[****(x _{2} - x_{1})^{2} + (y_{2} - y_{1})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 **P****1**** (1,2)** and **P****2**** (4,6)**. Find the distance d between **P****1** and **P****2.**

**Solution:**

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

Δx = x_{2} - x_{1}

Δx = 4 − 1

`Δx = 3`

**Step 2:** Square the difference obtained in **Step 1**.

(Δx)^{2} = 3^{2}

`(Δx)`

^{2} = 9

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

Δy = y_{2} - y_{1}

Δy = 6 − 2

`Δy = 4`

**Step 4:**** **Square the difference obtained in **Step 3**.

(Δy)^{2} = 4^{2}

`(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.