# Understanding the Distance Formula, its Derivation & How to Calculate Distance Between two Points?

In this article, you will learn the distance formula and how to calculate the distance between two points.

## Distance Formula

The term distance refers to the space between two objects. It tells that how far is the object from you. In geometry, the distance formula gives you the distance between two points or the distance between a point and a line or the distance between two parallel lines etc.

## What is the Distance formula?

The distance between two points can be calculated by taking the square root of the subtraction of x and y-coordinates. The distance formula between X and Y in two dimensions is given as:

## How to Derive Distance Formula?

The distance formula can be derived from the Pythagoras theorem for a right triangle. The formula for pythagoream theorem is stated as:

$$AB^2 \;=\; AC^2 \;+\; BC^2$$

Consider a line AB having points (x1,y1) and (x2,y2). Draw two lines from A to C and C to B by denoting the points in the xy-plane.

The distance between A and C along x-axis is;

$$AC \;=\; x_2 \;-\; x_1$$

The distance between A and C along x-axis is:

$$BC \;=\; y_2 \;-\; y_1$$

By using Pythagoras theorem,

$$d \;=\; (x_2 - x_1)^2 \;+\;(y_2 - y_1)^2$$

Here d is the d is the distance from A to B.

The distance formula can be obtained by taking square root of the above equation so,

$$d \;=\; \sqrt {(x_2 - x_1)^2 \;+\;(y_2 - y_1)^2}$$

Which is the distance between two points formula to calculate the distance between A and B. Let’s see an example to find the distance between two points.

Consider the point A has coordinates (2, 5) and B has coordinates (6,- 2) We will use distance formula for A and B which is:

$$d \;=\; \sqrt {(x_2 - x_1)^2 \;+\;(y_2 - y_1)^2}$$

Substituting the values of A and B,

$$d \;=\; \sqrt{(6-2)^2 \;+\; (-2-5)^2}$$ $$d \;=\; \sqrt{(4)^2 \;+\; (-7)^2}$$ $$d \;=\; \sqrt {16 \;+\; 49}$$ $$d \;=\; \sqrt 65$$

So the distance between A and B is √65. But instead of following manual steps, calculatored also offers free distance formula solver. This will reduces efforts of doing calculations for finding the distance between two points.

## How to Calculate Distance between Two Points?

The distance between two points can be calculated by using the distance formula. Consider two point of a line segment A and B having coordinates (a1,b1) and (a2,b2). The distance between A and B can be calculated by following formula:

$$d \;=\; \sqrt {(a_2 - a_1)^2 \;+\;(b_2 - b_1)^2}$$

Let’s see the following example to understand how the distance can be calculated between two points.

## Related Formulas to Distance Formula

There are some formulas which are related to distance formulas. These are:

1. Distance formula from a point to a line.
2. Distance formula for two parallel lines.
3. Distance formula for the distance between a point and a plane.
4. Midpoint formula
5. Slope formula

### Distance Formula from a Point to a Line

It is used to calculate the distance between a point and a line. The distance formula to calculate distance between a point p(x1,y1) and a line ax + by + c = 0 in two dimensions is:

$$d \;=\; \frac{|ax_1 + by_1 + c|}{ \sqrt{a^2+b^2}}$$

Where the equation of the line is ax + by + c = 0.

### Distance Formula for two Parallel Lines.

It is used to calculate the distance between two parallel lines in two dimensions. The formula for the distance between two lines A: ax1+by1+c1 = 0 and B:ax2+by2+c2=0 is:

$$d \;=\; \frac{|c_2-c_1|}{ \sqrt{a^2+b^2}}$$

### Distance Formula for the Distance between a Point and a Plane.

It is used to calculate the distance between a point and a plane in xyz-coordinates. The formula to calculate distance between a point (x1,y1,z1) and a plane ax+by+cz+d=0 is:

$$d\;=\; \frac{|ax_1+by_1+cz_1+d|}{ \sqrt{a^2+b^2+c^2}}$$

### Midpoint Formula

The midpoint formula is used to calculate the center point of any line segment formed by joining two points. The mathematical formula for calculating the midpoint is:

$$M \;=\; \Biggr( \frac{x_1+x_2}{2} \;,\; \frac{y_1+y_2}{2} \Biggr)$$

### Slope Formula

It is used to calculate the steepness of a line. The slope formula calculates the ratio of vertical distance to horizontal distance between two distinct points on a same line. Consider we have two points (x1,y1) and (x2,y2). So the formula of steepness is:

$$m \;=\; \frac{y_2 - y_1}{x_2 - x_1}$$

## What is the Distance Formula?

The distance formula between X and Y in two dimensions is given as:

$$d \;=\; \sqrt {(x_2 - x_1)^2 \;+\;(y_2 - y_1)^2}$$

## What is Distance in Math?

The distance in math is the distance between two points on a line segment. It is also referred to as the distance between two parallel lines or the distance between a point and a line or the distance between a point and a plane

### Shaun Murphy

Last updated: July 07, 2023

I'm graduated in biomedical and electrical engineering. Specialization in sports and medical topics but will gladly tackle everything you throw at him. He is a sailor, hiker, and motorcyclist in his free time.