Distance Between Two Points Formula

The Distance Between Two Points Formula is one of the fundamental tools in coordinate geometry. It allows us to calculate the straight-line distance between two points located in either two-dimensional or three-dimensional space. Whether you're preparing for exams or using a distance between two points calculator for quick solutions, this formula plays a critical role in mathematics, engineering, and physics.

Let’s explore the derivation, formula, and real-world application of this concept.

 

Table of Contents

• What is the Distance Between Two Points Formula?
• Understanding the Distance Formula with Coordinates
• Distance Between Two Points on a Coordinate Plane
• Distance Between Two Points Formula in 3D Space
• Solved Examples
• When to Use the Distance Between Two Points Calculator?
• Conclusion
• FAQs on Distance Between Two Points Formula

What is the Distance Between Two Points Formula?

In geometry, the distance between two points refers to the length of the shortest straight line that joins them. When the coordinates of the points are known, the distance can be calculated using a formula derived from the Pythagorean Theorem.

Formula (2D Coordinate Plane):

If A(x₁, y₁) and B(x₂, y₂) are two points in a 2D plane,
then the distance between them is given by:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

This formula is applied in most distance between two points calculator tools and helps in solving problems involving line segments, graphs, and shapes in the coordinate plane.

 

Understanding the Distance Formula with Coordinates

The Distance Between Two Points Formula can be applied in different coordinate arrangements:

Points on the x-axis:
If the points are (x₁, 0) and (x₂, 0), then:
d = |x₂ − x₁|

Points on the y-axis:
If the points are (0, y₁) and (0, y₂), then:
d = |y₂ − y₁|

These are simplified cases of the general formula used in 2D geometry.

 

Distance Between Two Points on a Coordinate Plane

In a two-dimensional (x-y) coordinate system, any point is defined by:

• x-coordinate (abscissa): Its horizontal distance from the y-axis
  
  

• y-coordinate (ordinate): Its vertical distance from the x-axis
  
  

To find the distance from the origin (0, 0) to any point P(x, y), the formula becomes:

d = √(x² + y²)

This is a special case where one of the points is at the origin, simplifying the general formula.

 

Distance Between Two Points Formula in 3D Space

In three-dimensional space, a z-coordinate is added to represent depth or height. The Distance Between Two Points Formula expands as:

If A(x₁, y₁, z₁) and B(x₂, y₂, z₂), then:
d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]

To calculate the distance of a point P(x, y, z) from the origin O(0, 0, 0), use:
d = √(x² + y² + z²)

This is often applied in fields like 3D modeling, architecture, and physics.

 

Solved Examples

Example 1: Find the Distance Between Two Points

 Q: What is the distance between A(3, 2) and B(9, 7)?
Solution:
Using the formula:
d = √[(9 − 3)² + (7 − 2)²]
= √(36 + 25)
= √61
Answer: d ≈ 7.81 units

 

Example 2: Find the Distance from Origin

 Q: Find the distance of point P(4, 3) from the origin.
Solution:
Using: d = √(x² + y²)
= √(16 + 9)
= √25
Answer: d = 5 units

 

When to Use the Distance Between Two Points Calculator?

Using a distance between two points calculator is helpful when:

• You're working with large or decimal coordinates

• Solving multiple geometry problems quickly

• Working in three dimensions

• Verifying your manual calculations
  
  

These calculators simplify the task by directly applying the formula and delivering the result.

 

Conclusion

The Distance Between Two Points Formula is a cornerstone concept in coordinate geometry. It helps calculate the shortest path between two locations in both 2D and 3D spaces. Whether done manually or using a distance between two points calculator, knowing this formula enhances problem-solving skills and lays the foundation for advanced geometric applications.

 

Frequently Asked Questions On Distance Between Two Points Formula

1. How to calculate the distance between two points?

Answer.To calculate the distance between two points, you use the Distance Formula.
If the two points are A(x₁, y₁) and B(x₂, y₂) on a 2D coordinate plane, the formula is:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

This formula comes directly from the Pythagorean Theorem. You simply find the difference in x-coordinates and y-coordinates, square them, add them, and then take the square root.

 2.What is the distance between (3, 10) and (3, −8) on a coordinate grid?

Answer.Here, both points have the same x-coordinate (3). So they lie on a vertical line parallel to the y-axis.

Using the formula:
d = |y₂ − y₁| = |−8 − 10| = |−18| = 18

Ans: The distance is 18 units.

 3. How do you prove the distance between two points?

Answer.The proof comes from the Pythagorean Theorem:

• Imagine two points A(x₁, y₁) and B(x₂, y₂).

• Draw a right-angled triangle by dropping perpendiculars from the points to form horizontal and vertical lines.

• The difference in the x-coordinates, (x₂ − x₁), becomes the base of the triangle.

• The difference in the y-coordinates, (y₂ − y₁), becomes the height of the triangle.

• By the Pythagorean theorem:
  AB² = (x₂ − x₁)² + (y₂ − y₁)²

• Taking the square root, we get the distance formula:
  AB = √((x₂ − x₁)² + (y₂ − y₁)²)

 4. How do you find the distance between two points in space?

Answer.In 3D space, each point has three coordinates: (x, y, z).

If the two points are A(x₁, y₁, z₁) and B(x₂, y₂, z₂), then the distance is:
d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²)

This is just an extension of the 2D formula, with an extra term for the z-axis (depth/height).

 

Explore the concept in depth and practice with Orchids International to build confidence in using the distance formula accurately.