Distance Formula
Calculate the distance between two points in a coordinate plane using the distance formula d = √((x₂-x₁)² + (y₂-y₁)²).
Includes step-by-step examples.
The Formula
d = √((x₂ - x₁)² + (y₂ - y₁)²)
The distance formula calculates the straight-line distance between two points on a coordinate plane.
It is derived directly from the Pythagorean theorem.
Variables
| Symbol | Meaning |
|---|---|
| d | The distance between the two points |
| (x₁, y₁) | Coordinates of the first point |
| (x₂, y₂) | Coordinates of the second point |
Example 1
Find the distance between (1, 2) and (4, 6)
x₁ = 1, y₁ = 2, x₂ = 4, y₂ = 6
d = √((4 - 1)² + (6 - 2)²)
d = √(3² + 4²) = √(9 + 16) = √25
d = 5
Example 2
Find the distance between (-3, 7) and (2, -1)
x₁ = -3, y₁ = 7, x₂ = 2, y₂ = -1
d = √((2 - (-3))² + (-1 - 7)²)
d = √(5² + (-8)²) = √(25 + 64) = √89
d ≈ 9.434
When to Use It
Use the distance formula when:
- You need the straight-line distance between two points on a graph
- Checking if a point lies on a circle (compare distance to the radius)
- Solving geometry problems involving coordinate planes
- Calculating lengths of line segments in analytic geometry
Key Notes
- The formula is the Pythagorean theorem applied to the right triangle formed by the horizontal run (x₂−x₁) and vertical rise (y₂−y₁) — drawing this triangle is the best way to visualize why it works and catch errors
- Order of subtraction never matters because both differences are squared: (x₂−x₁)² = (x₁−x₂)², so it makes no difference which point is "1" or "2" — a common worry that adds no actual constraint
- Extends to 3D naturally: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²); in n dimensions it becomes the Euclidean norm, which underpins k-nearest-neighbor classifiers, clustering algorithms, and vector similarity in machine learning
- Do not use the distance formula for geographic (lat/lon) coordinates — the Earth is curved, so straight-line 2D distance is incorrect; use the Haversine formula or great-circle distance for accurate surface measurements