Distance Formula Calculator
Coordinates
Enter the coordinates of two points to find the distance between them.
Understanding the Distance Formula
The distance formula is used to determine the length of the line segment connecting two points in a Euclidean plane. It is derived directly from the Pythagorean theorem.
Given two points, Point 1 (x₁, y₁) and Point 2 (x₂, y₂), the distance d is calculated as:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]How it Works
Imagine a right-angled triangle where the hypotenuse is the line segment connecting our two points.
- The horizontal leg has length |x₂ - x₁| (change in x).
- The vertical leg has length |y₂ - y₁| (change in y).
- According to Pythagoras (a² + b² = c²), the squared distance is the sum of the squared legs.
- Taking the square root gives the final distance.
Examples
Example 1
Find the distance between (1, 2) and (4, 6).
- x₂ - x₁ = 4 - 1 = 3
- y₂ - y₁ = 6 - 2 = 4
- d = √(3² + 4²) = √(9 + 16) = √25 = 5
Example 2
Find the distance between (-3, -1) and (2, 3).
- x₂ - x₁ = 2 - (-3) = 5
- y₂ - y₁ = 3 - (-1) = 4
- d = √(5² + 4²) = √(25 + 16) = √41 ≈ 6.403
Frequently Asked Questions
Can distance be negative?
No. Distance is a scalar quantity representing a physical length, so it is always non-negative.
What refers to 1D distance?
On a simple number line, the distance between a and b is simply |b - a|.
Does this work for 3D points?
The concept is the same, but you add the z-coordinate: d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].