triangle inequality: meaning, definition, pronunciation and examples
C2Formal, Technical, Academic
Quick answer
What does “triangle inequality” mean?
A fundamental theorem in mathematics stating that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Audio
Pronunciation
Definition
Meaning and Definition
A fundamental theorem in mathematics stating that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
A principle or theorem in various mathematical fields (e.g., geometry, metric spaces, vector spaces) that generalises the geometric rule to abstract distances, stating that the distance from A to C is always less than or equal to the sum of the distance from A to B and the distance from B to C.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or semantic differences. Spelling of related words may follow regional conventions (e.g., 'metre' vs. 'meter' in a physics context).
Connotations
Identical technical connotations in both varieties.
Frequency
Equally low frequency in general language but standard and identical in technical/academic registers in both regions.
Grammar
How to Use “triangle inequality” in a Sentence
The triangle inequality [verb: holds/fails/applies] in this space.One can [verb: prove/derive/use] the triangle inequality.This metric satisfies the triangle inequality.Vocabulary
Collocations
Examples
Examples of “triangle inequality” in a Sentence
verb
British English
- The function does not triangle-inequality.
American English
- The function does not triangle-inequality.
adjective
British English
- (Not standard; used attributively) The triangle-inequality property is crucial.
American English
- (Not standard; used attributively) The triangle-inequality principle is fundamental.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core concept in mathematics, physics, and engineering courses; used in proofs and definitions.
Everyday
Extremely rare outside of educational or puzzle contexts.
Technical
Essential in pure mathematics (metric spaces, analysis), computer science (algorithms, clustering), and physics.
Vocabulary
Synonyms of “triangle inequality”
Strong
Neutral
Weak
Vocabulary
Antonyms of “triangle inequality”
Watch out
Common Mistakes When Using “triangle inequality”
- Misstating it as 'the sum of two sides is greater than the third' (must be *strictly greater than* for non-degenerate triangles in Euclidean geometry).
- Applying it to non-metric contexts where distance is not defined.
FAQ
Frequently Asked Questions
No. While it originates from and is named after the geometric property of triangles, it is generalised as a fundamental axiom for 'distance' in abstract mathematics, such as in metric spaces.
Yes. In the geometric case, equality occurs only for a 'degenerate' triangle where all three points lie on a straight line. In abstract metric spaces, equality holds when point B lies on a 'shortest path' between A and C.
It is essential for ensuring that the concept of 'distance' behaves in an intuitive way, preventing paradoxical results. It is a foundational tool in analysis, optimisation, and algorithm design (e.g., in clustering).
It is a corollary stating that |d(A, C) - d(B, C)| ≤ d(A, B). It gives a lower bound on the distance between two points, in contrast to the standard inequality's upper bound.
A fundamental theorem in mathematics stating that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Triangle inequality is usually formal, technical, academic in register.
Triangle inequality: in British English it is pronounced /ˈtraɪæŋɡl̩ ˌɪnɪˈkwɒlɪti/, and in American English it is pronounced /ˈtraɪæŋɡl̩ ˌɪnɪˈkwɑːləti/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “(None directly; the term itself is technical)”
Learning
Memory Aids
Mnemonic
Imagine a triangle as a detour: going directly from point A to point C (one side) is always shorter than going via point B (the other two sides combined).
Conceptual Metaphor
THE SHORTCUT METAPHOR: The direct path is never longer than a broken path. / A DETOUR IS LONGER.
Practice
Quiz
In which of the following contexts is the triangle inequality a defining property?