Distance Formula Calculator

Calculate the distance between two points (x₁,y₁) and (x₂,y₂) using the distance formula. Also find the midpoint, slope of the line segment, and equation of the line.

∑ Algebra📐 d = √((x₂-x₁)² + (y₂-y₁)²)🔢 Math
Point 1 x₁
Point 1 y₁
Point 2 x₂
Point 2 y₂
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Distance Formula Calculator—d = √((x₂-x₁)² + (y₂-y₁)²)units

Step-by-Step Examples

Example 1
Two Points

(1,2) and (4,6).

  • dx=3, dy=4
  • d=√(9+16)=√25=5
  • Midpoint: (2.5, 4)
  • Slope: 4/3
✓ d = 5
Example 2
Origin to Point

(0,0) to (3,4).

  • d=√(9+16)=5
  • Classic 3-4-5 right triangle
✓ d = 5 (3-4-5 triangle)
Example 3
Negative Coords

(−2,−3) to (4,5).

  • dx=6, dy=8
  • d=√(36+64)=√100=10
✓ d = 10

Real-World Applications

🗺️
Navigation
GPS distance between two coordinates.
📊
Data Science
Euclidean distance between data points in machine learning.
🎮
Game Development
Distance between player and enemy for collision detection.
📐
Geometry
Length of line segments in coordinate geometry proofs.

Common Mistakes to Avoid

⚠️
Not squaring before adding

d = √((x2−x1)² + (y2−y1)²). Both differences must be squared before adding under the radical.

⚠️
Negative distance

Distance is always non-negative. If you get a negative value, check your arithmetic — the square root ensures positive results.

⚠️
Confusing distance with displacement

Distance is the magnitude between two points. Displacement is directional. Distance formula gives scalar distance, not direction.

Frequently Asked Questions

What is the distance formula? ▾
Derived from the Pythagorean theorem. The difference in x and y form legs of a right triangle; the distance is the hypotenuse.
How is it related to Pythagorean theorem? ▾
d² = (x₂−x₁)² + (y₂−y₁)² is exactly a² + b² = c², where a=Δx and b=Δy.
What is the midpoint formula? ▾
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2). The average of the x-coordinates and the average of the y-coordinates.
How do I find distance in 3D? ▾
d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). Same principle extended to three dimensions.
What is Manhattan distance? ▾
|x₂−x₁| + |y₂−y₁|. Used in city block navigation and some machine learning algorithms (L1 norm vs L2 norm).

Related Math Calculators

∑ Midpoint Formula∑ Slope∑ Pythagorean Theorem∑ Coordinate Geometry∑ Vector

Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

Double Angle Formula Calculator →Heron's Formula Calculator →Inverse Trigonometric Functions Calculator →Math Formula Explorer →