A metric on a set of objects defines a measure of distance on the set, that is, for each pair of objects it defines the distance between those objects. Formally a metric is a function \(d\) from pairs of elements of the set to the non-negative real numbers satisfying,

  1. \(d(a,b) = 0\) if and only if \(a=b\),
  2. \(d(a,b)=d(b,a)\), and
  3. \(d(a,b)+d(b,c)\geq d(a,c)\)

for all \(a\), \(b\), and \(c\) in the set. The third condition is known as the triangle inequality, and captures the idea that you can't draw a triangle with three lengths if one is longer than the other two combined.

The distance formula in the Cartesian plane is an example of a metric, but there are many other possible metrics on the Cartesian plane, and in more abstract settings the concept of a metric is central, for instance in vector spaces and Hilbert spaces.

Citation Info

  • [MLA] “metric.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 28 Mar 2013. Web. 28 Mar 2013. <>
  • [APA] metric (28 Mar 2013). Retrieved 28 Mar 2013 from the Platonic Realms Interactive Mathematics Encyclopedia:


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