Distance Calculator

Calculate the distance between two points using Euclidean, Manhattan, or Haversine formulas. Supports 2D, 3D, and geographic coordinates.

Distance Result

Distance

5.0000

units

Calculate Distance

Point 1 (x₁, y₁)

Point 2 (x₂, y₂)

Formula & Calculation

Formula

d = √[(x₂-x₁)² + (y₂-y₁)²]

Distance

5.000000

units

Step-by-Step Solution

1.Δx = 4 - 1 = 3
2.Δy = 6 - 2 = 4
3.d = √(3² + 4²)
4.d = √(9 + 16)
5.d = √25
6.d = 5.000000

Visualization

xy(1, 2)(4, 6)
Point 1 Point 2 Distance

Distance Types Explained

Euclidean Distance

"As the crow flies" - straight-line distance. Uses Pythagorean theorem. Most common for geometric calculations.

Manhattan Distance

Sum of absolute differences in each dimension. Used in grid-based pathfinding, machine learning, and urban navigation.

Haversine Formula

Great-circle distance on a sphere. Accounts for Earth's curvature. Essential for GPS/navigation applications.

3D Distance

Extended Euclidean formula for 3D space. Used in 3D graphics, physics simulations, and spatial analysis.

Distance Result

Distance

5.0000

units

?How Do You Calculate Distance?

The distance formula finds the straight-line distance between two points. In 2D: d = sqrt((x2-x1)^2 + (y2-y1)^2). In 3D: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2). This is derived from the Pythagorean theorem. For points (1,2) and (4,6), distance = sqrt(9+16) = sqrt(25) = 5.

What is the Distance Formula?

Distance in coordinate geometry is the length of the straight line connecting two points. The distance formula, derived from the Pythagorean theorem, calculates this Euclidean distance using the differences in coordinates. It is fundamental to geometry, physics, navigation, computer graphics, and many scientific applications.

Key Facts About Distance Calculations

  • 2D distance: d = sqrt((x2-x1)^2 + (y2-y1)^2)
  • 3D distance: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
  • Based on Pythagorean theorem (a^2 + b^2 = c^2)
  • Also called Euclidean distance (straight line)
  • Manhattan distance uses |x2-x1| + |y2-y1| (grid paths)
  • Distance is always positive (absolute value)
  • Midpoint formula: ((x1+x2)/2, (y1+y2)/2)
  • Used in physics, navigation, graphics, and machine learning

Quick Answer

The distance formula finds the straight-line distance between two points. In 2D: d = sqrt((x2-x1)^2 + (y2-y1)^2). In 3D: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2). This is derived from the Pythagorean theorem. For points (1,2) and (4,6), distance = sqrt(9+16) = sqrt(25) = 5.

Frequently Asked Questions

The distance formula finds the length between two points. In 2D: d = √[(x₂-x₁)² + (y₂-y₁)²]. This is derived from the Pythagorean theorem. In 3D, add (z₂-z₁)² under the square root.
Manhattan distance (also called taxicab or L1 distance) measures distance along grid lines, like walking city blocks. It's |x₂-x₁| + |y₂-y₁|. Named after Manhattan's grid street layout where you can't travel diagonally.
Use the Haversine formula for great-circle distance on a sphere. It accounts for Earth's curvature. For short distances (<10km), the simple formula works well. For accuracy, use WGS84 ellipsoid calculations.
Use 2D Euclidean for flat surfaces and direct "as the crow flies" distance. Use 3D for points in space. Use Manhattan for grid-based movement (game AI, urban navigation). Use Haversine for geographic coordinates on Earth.
Distance is the total length traveled along any path (scalar, always positive). Displacement is the shortest straight-line distance between start and end points with direction (vector). Distance ≥ |Displacement|.

Last updated: 2025-01-15