Coordinate Distance Calculator

What is the Distance Formula?

The distance formula calculates the straight-line (Euclidean) distance between two points in a coordinate system. It is derived from the Pythagorean theorem and is fundamental to geometry, physics, and many other fields.

Distance Formulas

2D Distance Formula

For two points (x1, y1) and (x2, y2) in a 2D plane:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

3D Distance Formula

For two points (x1, y1, z1) and (x2, y2, z2) in 3D space:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Understanding the Formula

The distance formula is an extension of the Pythagorean theorem:

• In 2D, imagine a right triangle where the distance is the hypotenuse
• The horizontal leg has length |x2 - x1|
• The vertical leg has length |y2 - y1|
• The distance is sqrt(horizontal^2 + vertical^2)

How to Use This Calculator

1. Select 2D or 3D mode
2. Enter the coordinates for Point 1
3. Enter the coordinates for Point 2
4. Click "Calculate" to get the distance

Example Calculations

Example 1: 2D Distance

Find the distance between points (1, 2) and (4, 6):

d = sqrt((4 - 1)^2 + (6 - 2)^2)
d = sqrt(3^2 + 4^2)
d = sqrt(9 + 16)
d = sqrt(25)
d = 5

Example 2: 3D Distance

Find the distance between points (1, 2, 3) and (4, 6, 3):

d = sqrt((4 - 1)^2 + (6 - 2)^2 + (3 - 3)^2)
d = sqrt(3^2 + 4^2 + 0^2)
d = sqrt(9 + 16 + 0)
d = sqrt(25)
d = 5

Example 3: Diagonal of a Room

A room is 12 feet long, 9 feet wide, and 8 feet tall. Find the diagonal distance from one corner to the opposite corner:

d = sqrt(12^2 + 9^2 + 8^2)
d = sqrt(144 + 81 + 64)
d = sqrt(289)
d = 17 feet

Applications

Navigation and GPS

GPS systems use distance calculations to determine routes and estimate travel times.

Computer Graphics

Collision detection, pathfinding, and proximity calculations all rely on distance formulas.

Physics

Calculating displacement, electric field strength at a distance, and gravitational forces.

Data Science

K-nearest neighbors algorithms and clustering use Euclidean distance to measure similarity between data points.

Related Concepts

• Manhattan Distance: Sum of absolute differences (|x2-x1| + |y2-y1|)
• Chebyshev Distance: Maximum of absolute differences
• Minkowski Distance: Generalized distance formula