What is the distance formula in maths?

The distance formula in maths finds the distance between two points in an XY plane.

Where is the distance formula in maths derived from?

The distance formula in maths is derived from the Pythagorean theorem.

What are some real-life applications of the distance formula in maths?

A distance formula in maths is typically used in real-life scenarios like navigation systems, computer graphics, and sports analytics.

What are some things to keep in mind while you calculate the distance formula?

While you calculate distance formula, one should remember to check the coordinates carefully before making the calculations. One should also always compute the square differences first before adding. Some square roots may not be exact, so you would need to use approximations when needed.

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Distance Formula

1.0Master Coordinate Distances in Minutes

Learn how to calculate the exact straight-line distance between any two points on a two-dimensional grid. Master the coordinate system, understand the geometric derivation using the Pythagorean theorem, and apply the distance formula to solve algebraic and geometric board exam problems—all formatted in clean, plain text for simple copy-pasting.

Class: 10 Mathematics (CBSE)
Chapter: Coordinate Geometry
Estimated Learning Time: 15–20 Minutes

2.0Learning Outcomes

After completing this lesson, you will be able to:

Identify points on a Cartesian plane using x-coordinates (abscissa) and y-coordinates (ordinate).
State and derive the Distance Formula using the Pythagorean theorem.
Compute the distance of any point from the origin (0, 0).
Apply the distance formula to check geometric properties like collinearity of points and types of triangles or quadrilaterals.

3.0What is the Distance Formula?

To calculate distance formula between two points (x₁,y₁) and (x₂,y₂) in a two-dimensional plane, you have to:

$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

Here,

D is the distance between two points.
(x₁,y₁) and (x₂,y₂) are the coordinates of the points.

4.0Derivation of the Distance Formula

The distance formula geometry is based on the Pythagorean theorem, which states:

a²+ b²= c²

Here, c represents the length of the hypotenuse in a right triangle; a and b are the other sides.

Considering the two points A (x₁,y₁) and B(x₂,y₂), we can form a right triangle by drawing a horizontal and a vertical line from these two points. The horizontal distance (x₂- x₁) represents one leg of the triangle, and the vertical difference (y₂-y₁) represents the other leg. So, applying the Pythagorean theorem, we get:

$D^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

5.0Distance Formula Example

Let’s take distance formula examples and solve them so you can grasp the concept better.

Example: Find the distance between the points A (2,3) and B (6,7)

Solution: Using the distance formula in Maths, we can see,

$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} D = (6 - 2)^{2} + (7 - 3)^{2} D = (4)^{2} + (4)^{2} D = 32 D \approx 5.66$

Thus, the distance between the two points is 5.66 units.

6.0Distance Formula in Different Coordinate Systems

Refer to the table below for the distance formula geometry in different coordinate systems:

Coordinate System	Distance Formula
2D Cartesian Plane	$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
3D Cartesian Plane	$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$
Polar Coordinates	$D = r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} Cos θ$

7.0Distance Formula Problems

To master the concept of distance formula, one needs to look closely at distance formula examples with answers. Let’s look at some distance formula problems to understand how to apply the concepts mathematically and in real-life situations.

Problem 1: Find the distance between the points (4, –2) and (–3, 5).

Solution:

$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} D = (- 3 - 4)^{2} + (5 - (- 2))^{2} D = (- 7)^{2} + (7)^{2} D = 49 + 49 = 98 D \approx 9.90$

Problem 2: The distance between the points A(4, 5) and B(x, 9) is 5 units. Find the possible values of x.

Solution: Let A(4, 5) be A(x₁,y₁), and B(x, 9) be B(x₂,y₂). Now according to the question:

$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} 5 = (x - 4)^{2} + (9 - 5)^{2} 5 = (x - 4)^{2} + (4)^{2}$

Squaring both sides:

25 = x²+16–8x +16

x² – 8x + 7 = 0

x² – 7x – x +7 =0

x(x-7) – 1(x–7)=0

(x – 1)(x–7) = 0

x = 1, 7

8.0Distance Formula Problem-Solving Tips

While doing distance formula problem-solving, you should keep in mind the following tips and tricks to efficiently master the exercises.

Plot the Points: Visual aids work better than just numbers. You can always plot the points in a graph to understand the problem better.
Check Coordinates Carefully: Check the coordinates carefully before making the calculations. Small mistakes can lead to an incorrect result in the end.
Square Each Difference First: Always compute the square difference first before adding.
Use Approximation When Needed: Some square roots may not be exact, so you should round off accordingly.

9.0Applications of the Distance Formula

The distance formula in math is an important concept that has crucial real-life applications.

Navigation Systems: It is used in GPS technology to calculate the distance between two locations.
Physics: Helps in motion analysis and speed calculations.
Computer Graphics: It is essential for rendering objects in a three-dimensional space.
Sports Analytics: It is used in various sports to understand player movements and ball trajectories.

10.0EUREKA by ALLEN – The Future of Class 10 Learning

With EUREKA by ALLEN, you're going to change the way you learn as a Class 10 student using an AI-based educational platform designed just for you! EUREKA offers everything from online videos, customized learning paths, immediate question resolution, and live statistics about how well you are doing with ALLEN's educational professional discipline combined with advanced technology for a more intelligent and effective learning experience!

Key Features of EUREKA Class 10 Courses:
Interactive concept learning modules Engaging story-based video lectures CBSE & State Board exam preparation Subjective answer writing practice Instant quizzes with performance feedback Detailed progress tracking 24×7 AI-powered doubt assistance NCERT-aligned curriculum Learn anytime, anywhere

Explore Now

11.0Supporting Study Materials

This study material CBSE Notes and NCERT Solutions for the Chapter "Coordinate Geometry" on Distance Formula Topics is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It provides clear explanations of key concepts, definitions, formulas, and important questions to help students understand the distance between two points in a Cartesian plane, collinearity of points, geometric properties of triangles and quadrilaterals, and prepare effectively for examinations.

CBSE Class 10 Maths Notes Chapter 7 Coordinate Geometry	NCERT Solutions for Class 10 Maths Chapter 7: Coordinate Geometry

12.0Previous Year Question on Distance Formula

Problem: Three vertices of a triangle are given as A(0, 0), B(6, 0), and C(6, 8). Prove whether it is a right-angled triangle.

Solution: Let’s use distance formula $(D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2})$ to prove if the vertices belong to a right-angled triangle or not.

$AB = (6 - 0)^{2} + (0 - 0)^{2} = 6 BC = (6 - 6)^{2} + (8 - 0)^{2} = 8 AC = (6 - 0)^{2} + (8 - 0)^{2} = 100 = 10$

Now, applying Pythagoras Theorem: Check if AC² = AB² + BC²

AC² = (10)² = 100

AB² + BC² = 6^{2 .}+ 8² = 36 + 64 = 100

Here, AC² = AB² + BC², hence ABC is a right-angled triangle.

13.030-Second Revision: Distance Formula

Distance Formula: Used to find the length between (x1, y1) and (x2, y2).
Equation: $D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

From Origin (0,0): $D = (x)^{2} + (y)^{2}$

Collinear Points: Three points are on a line if AB + BC = AC.
Square vs Rhombus: Both have 4 equal sides, but only a square has equal diagonals (AC = BD).
Rectangle vs Parallelogram: Both have equal opposite sides, but only a rectangle has equal diagonals.
14.0Recommended Next Topics
Section Formula
Similarity Criterion
Basic Proportionality Theorem (BPT)
Trigonometric Identities

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Distance Formula

1.0Master Coordinate Distances in Minutes

Learn how to calculate the exact straight-line distance between any two points on a two-dimensional grid. Master the coordinate system, understand the geometric derivation using the Pythagorean theorem, and apply the distance formula to solve algebraic and geometric board exam problems—all formatted in clean, plain text for simple copy-pasting.

Class: 10 Mathematics (CBSE)
Chapter: Coordinate Geometry
Estimated Learning Time: 15–20 Minutes

2.0Learning Outcomes

After completing this lesson, you will be able to:

Identify points on a Cartesian plane using x-coordinates (abscissa) and y-coordinates (ordinate).
State and derive the Distance Formula using the Pythagorean theorem.
Compute the distance of any point from the origin (0, 0).
Apply the distance formula to check geometric properties like collinearity of points and types of triangles or quadrilaterals.

3.0What is the Distance Formula?

To calculate distance formula between two points (x₁,y₁) and (x₂,y₂) in a two-dimensional plane, you have to:

$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

Here,

D is the distance between two points.
(x₁,y₁) and (x₂,y₂) are the coordinates of the points.

4.0Derivation of the Distance Formula

The distance formula geometry is based on the Pythagorean theorem, which states:

a²+ b²= c²

Here, c represents the length of the hypotenuse in a right triangle; a and b are the other sides.

Considering the two points A (x₁,y₁) and B(x₂,y₂), we can form a right triangle by drawing a horizontal and a vertical line from these two points. The horizontal distance (x₂- x₁) represents one leg of the triangle, and the vertical difference (y₂-y₁) represents the other leg. So, applying the Pythagorean theorem, we get:

$D^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

5.0Distance Formula Example

Let’s take distance formula examples and solve them so you can grasp the concept better.

Example: Find the distance between the points A (2,3) and B (6,7)

Solution: Using the distance formula in Maths, we can see,

$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} D = (6 - 2)^{2} + (7 - 3)^{2} D = (4)^{2} + (4)^{2} D = 32 D \approx 5.66$

Thus, the distance between the two points is 5.66 units.

6.0Distance Formula in Different Coordinate Systems

Refer to the table below for the distance formula geometry in different coordinate systems:

Coordinate System	Distance Formula
2D Cartesian Plane	$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
3D Cartesian Plane	$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}$
Polar Coordinates	$D = r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} Cos θ$

7.0Distance Formula Problems

To master the concept of distance formula, one needs to look closely at distance formula examples with answers. Let’s look at some distance formula problems to understand how to apply the concepts mathematically and in real-life situations.

Problem 1: Find the distance between the points (4, –2) and (–3, 5).

Solution:

$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} D = (- 3 - 4)^{2} + (5 - (- 2))^{2} D = (- 7)^{2} + (7)^{2} D = 49 + 49 = 98 D \approx 9.90$

Problem 2: The distance between the points A(4, 5) and B(x, 9) is 5 units. Find the possible values of x.

Solution: Let A(4, 5) be A(x₁,y₁), and B(x, 9) be B(x₂,y₂). Now according to the question:

$D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} 5 = (x - 4)^{2} + (9 - 5)^{2} 5 = (x - 4)^{2} + (4)^{2}$

Squaring both sides:

25 = x²+16–8x +16

x² – 8x + 7 = 0

x² – 7x – x +7 =0

x(x-7) – 1(x–7)=0

(x – 1)(x–7) = 0

x = 1, 7

8.0Distance Formula Problem-Solving Tips

While doing distance formula problem-solving, you should keep in mind the following tips and tricks to efficiently master the exercises.

Plot the Points: Visual aids work better than just numbers. You can always plot the points in a graph to understand the problem better.
Check Coordinates Carefully: Check the coordinates carefully before making the calculations. Small mistakes can lead to an incorrect result in the end.
Square Each Difference First: Always compute the square difference first before adding.
Use Approximation When Needed: Some square roots may not be exact, so you should round off accordingly.

9.0Applications of the Distance Formula

The distance formula in math is an important concept that has crucial real-life applications.

Navigation Systems: It is used in GPS technology to calculate the distance between two locations.
Physics: Helps in motion analysis and speed calculations.
Computer Graphics: It is essential for rendering objects in a three-dimensional space.
Sports Analytics: It is used in various sports to understand player movements and ball trajectories.

10.0EUREKA by ALLEN – The Future of Class 10 Learning

With EUREKA by ALLEN, you're going to change the way you learn as a Class 10 student using an AI-based educational platform designed just for you! EUREKA offers everything from online videos, customized learning paths, immediate question resolution, and live statistics about how well you are doing with ALLEN's educational professional discipline combined with advanced technology for a more intelligent and effective learning experience!

Key Features of EUREKA Class 10 Courses:
Interactive concept learning modules Engaging story-based video lectures CBSE & State Board exam preparation Subjective answer writing practice Instant quizzes with performance feedback Detailed progress tracking 24×7 AI-powered doubt assistance NCERT-aligned curriculum Learn anytime, anywhere

Explore Now

11.0Supporting Study Materials

This study material CBSE Notes and NCERT Solutions for the Chapter "Coordinate Geometry" on Distance Formula Topics is designed according to the latest CBSE Class 10 Mathematics syllabus and NCERT guidelines. It provides clear explanations of key concepts, definitions, formulas, and important questions to help students understand the distance between two points in a Cartesian plane, collinearity of points, geometric properties of triangles and quadrilaterals, and prepare effectively for examinations.

CBSE Class 10 Maths Notes Chapter 7 Coordinate Geometry	NCERT Solutions for Class 10 Maths Chapter 7: Coordinate Geometry

12.0Previous Year Question on Distance Formula

Problem: Three vertices of a triangle are given as A(0, 0), B(6, 0), and C(6, 8). Prove whether it is a right-angled triangle.

Solution: Let’s use distance formula $(D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2})$ to prove if the vertices belong to a right-angled triangle or not.

$AB = (6 - 0)^{2} + (0 - 0)^{2} = 6 BC = (6 - 6)^{2} + (8 - 0)^{2} = 8 AC = (6 - 0)^{2} + (8 - 0)^{2} = 100 = 10$

Now, applying Pythagoras Theorem: Check if AC² = AB² + BC²

AC² = (10)² = 100

AB² + BC² = 6^{2 .}+ 8² = 36 + 64 = 100

Here, AC² = AB² + BC², hence ABC is a right-angled triangle.

13.030-Second Revision: Distance Formula

Distance Formula: Used to find the length between (x1, y1) and (x2, y2).
Equation: $D = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

From Origin (0,0): $D = (x)^{2} + (y)^{2}$

Collinear Points: Three points are on a line if AB + BC = AC.
Square vs Rhombus: Both have 4 equal sides, but only a square has equal diagonals (AC = BD).
Rectangle vs Parallelogram: Both have equal opposite sides, but only a rectangle has equal diagonals.
14.0Recommended Next Topics
Section Formula
Similarity Criterion
Basic Proportionality Theorem (BPT)
Trigonometric Identities

Frequently Asked Questions

What is the distance formula in maths?

Where is the distance formula in maths derived from?

What are some real-life applications of the distance formula in maths?

What are some things to keep in mind while you calculate the distance formula?

Frequently Asked Questions

What is the distance formula in maths?

Where is the distance formula in maths derived from?

What are some real-life applications of the distance formula in maths?

What are some things to keep in mind while you calculate the distance formula?

Join ALLEN!

Distance Formula

1.0Master Coordinate Distances in Minutes

2.0Learning Outcomes

3.0What is the Distance Formula?

4.0Derivation of the Distance Formula

5.0Distance Formula Example

6.0Distance Formula in Different Coordinate Systems

7.0Distance Formula Problems

8.0Distance Formula Problem-Solving Tips

9.0Applications of the Distance Formula

10.0EUREKA by ALLEN – The Future of Class 10 Learning

11.0Supporting Study Materials

12.0Previous Year Question on Distance Formula

13.030-Second Revision: Distance Formula

14.0Recommended Next Topics

Table of Contents

Join ALLEN!

Distance Formula

1.0Master Coordinate Distances in Minutes

2.0Learning Outcomes

3.0What is the Distance Formula?

4.0Derivation of the Distance Formula

5.0Distance Formula Example

6.0Distance Formula in Different Coordinate Systems

7.0Distance Formula Problems

8.0Distance Formula Problem-Solving Tips

9.0Applications of the Distance Formula

10.0EUREKA by ALLEN – The Future of Class 10 Learning

11.0Supporting Study Materials

12.0Previous Year Question on Distance Formula

13.030-Second Revision: Distance Formula

14.0Recommended Next Topics

Table of Contents