Distance Between Two Points: Formula, Steps & Solved Examples

Reviewed by:

Rama Sharma

How to Find the Distance Between Two Points in Maths?

The concept of Distance Between Two Points plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Whether you're calculating length on a coordinate graph or finding the shortest path between two cities on a map, this formula is an essential tool for students and professionals alike.

What Is Distance Between Two Points?

The distance between two points is defined as the shortest straight-line measurement connecting those two points, either in a 2D coordinate plane or in 3D space. You’ll find this concept applied in areas such as coordinate geometry, mapping locations, and physics problems involving straight-line motion.

Key Formula for Distance Between Two Points

Here’s the standard formula:
\( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
For three dimensions, the formula extends to:
\( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)

Cross-Disciplinary Usage

Distance between two points is not only useful in Maths but also plays an important role in Physics (motion, displacement), Computer Science (algorithms, graphics), and daily logical reasoning (finding routes). Students preparing for exams like JEE, NEET, or school Olympiads will see its relevance in various questions, often combined with topics such as Coordinate Geometry or the Pythagorean Theorem.

Step-by-Step Illustration

Let's calculate the distance between the points A(2, 3) and B(-2, 0) using the 2D formula:

1. Identify the coordinates: \((x_1, y_1) = (2, 3)\) and \((x_2, y_2) = (-2, 0)\)

2. Plug values into the formula:
\( \text{Distance} = \sqrt{(-2 - 2)^2 + (0 - 3)^2} \)

3. Simplify inside the brackets:
First difference: -2 - 2 = -4
Second difference: 0 - 3 = -3

4. Square the differences:
(-4)² = 16
(-3)² = 9

5. Add the squares and take the square root:
\( \text{Distance} = \sqrt{16 + 9} = \sqrt{25} \)

6. Final answer: \( 5 \) units

Speed Trick or Vedic Shortcut

Here’s a quick tip—if two points have the same x or y coordinate, you can skip the whole formula! For example, if A(4, 0) and B(14, 0), the distance is just |14 - 4| = 10 units. This helps during timed exams and mental math. Tricks like these are often shared by Vedantu teachers in live classes to save students from unnecessary calculations.

Try These Yourself

Find the distance between (1, 2) and (4, 6).
Calculate the straight-line distance between (-3, 5) and (3, -1).
What is the distance between (0, 0, 0) and (1, 2, 2) in 3D?
On a graph, plot (5, 8) and (2, 4). How far apart are they?

Frequent Errors and Misunderstandings

Forgetting to square the differences before adding them.
Missing the minus sign: Always subtract in the same order for both x and y.
Not using absolute values: Distance is always positive.
Confusing 2D and 3D formulas—remember to include z in 3D cases!

Relation to Other Concepts

The idea of distance between two points connects closely with topics such as Midpoint of a Line Segment, Straight Lines, and Distance Between Two Parallel Lines. Mastering this concept helps in understanding slope, line equations, and more advanced geometry or coordinate problems.

Classroom Tip

A quick way to remember the distance formula is to see it as a real-life “Pythagoras shortcut”—if you draw a right triangle between two points on graph paper, the straight line (hypotenuse) is the actual distance. Many Vedantu teachers ask students to visualize a triangle for every pair of points to avoid mistakes.

We explored Distance Between Two Points—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept.