Distance is a basic concept that quantifies how far apart two points or objects are. In everyday language it denotes the amount of space between things; in physics and mathematics it is made precise by measurement procedures or by functions that assign a nonnegative number to a pair of points. Distance can be recorded along a path (the length of a route actually traveled) or as the shortest separation between positions, and those two ideas are not always the same. For example, walking once around a circular track produces a positive traveled distance while the straight-line separation between the start and end points can be zero; this contrast illustrates the difference between path length and net change in location.

Mathematical definitions and common formulas

In Euclidean geometry the straight-line distance between two points A and B in the plane is given by the Pythagorean relation: if A has coordinates (x1, y1) and B has (x2, y2), the distance is the square root of (x2−x1)^2+(y2−y1)^2. The Pythagorean theorem is an essential tool for that calculation and generalizes to higher dimensions. More abstractly, a distance function or metric on a set is any rule d(x,y) that satisfies certain properties: nonnegativity, identity of indiscernibles (d(x,y)=0 exactly when x=y), symmetry (d(x,y)=d(y,x)), and the triangle inequality (d(x,z) ≤ d(x,y)+d(y,z)). These axioms capture the intuitive behavior of separation and permit many alternatives to straight-line length.

Other distance measures

Different contexts require different notions of distance. Common alternatives include:

  • Manhattan (L1) distance: the sum of absolute coordinate differences, appropriate for grid paths.
  • Chebyshev (L∞) distance: the maximum coordinate difference, used in some optimization problems and chessboard moves.
  • Great-circle (geodesic) distance: the shortest path between two points on a sphere, used for long-distance navigation on Earth.
  • Graph distance: the minimum number of edges between nodes in a network.

Each of these alternatives still obeys the metric axioms (in standard formulations) but measures different features of separation that are useful in engineering, data analysis, and transport.

Measurement, units, and instruments

Practical measurement of distance depends on scale and accuracy requirements. Common units include metres, kilometres, feet and miles. Tools range from simple rulers and measuring tapes for short spans, to vehicle odometers, surveying chains, total stations, laser rangefinders, and satellite-based positioning systems (GPS) for longer or more precise measurements. In some settings the measured path length differs from the ideal mathematical distance: road distances follow road geometry, while map or coordinate distances represent projected separations.

History and development

Human societies have used distance measures since antiquity for trade, construction, and navigation, developing units tied to local standards. Formal study of distance entered mathematics with the geometry of the Greeks and was axiomatized later by metric-space theory in the 19th and 20th centuries. The Pythagorean relation, central to Euclidean distance, appears in many early engineering and surveying practices and remains foundational in science and technology.

Distinctions, examples, and notable facts

It is important to distinguish distance as a scalar quantity from displacement as a vector quantity: distance records how much ground has been covered regardless of direction, while displacement records the directed straight-line change in position. In physics the same word "distance" may refer to either path length or separation, so context is important — see displacement for the vector concept and scalar quantities for properties that have magnitude only. In geometry the term often appears linked to geometry and the notation for the segment between two points is standard; practical discussions of location refer to position and mapping methods. The basic computation in right-angled contexts uses the Pythagorean theorem, while more advanced or abstract treatments of separation look to the properties of a metric and to spaces where lines and shortest paths may be curved or constrained. For orientation in physics and applied contexts it helps to remember that a vector conveys direction and magnitude, whereas pure distance does not.

  1. Overview and everyday meaning of distance
  2. Mathematical metrics and Euclidean formulas
  3. Alternative measures and applications
  4. Measurement techniques and historical notes