The Triangle Inequality — Mathematics Knowledge System

Overview

The triangle inequality states that, for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the third side. Equivalently, three positive lengths a, b, c can form a triangle if and only if each one is less than the sum of the other two. The theorem captures a basic fact about straight-line distance: the direct path between two points is never longer than any path that detours through a third point. It underlies not just triangle geometry but the very definition of 'distance' (a metric) used throughout mathematics.

Intuition

Picture walking from town A to town C. Going directly is one route; going from A to B and then B to C is another. Detouring through B can never be shorter than going straight from A to C — at best (if B lies exactly on the direct path) it's the same length, and otherwise it's longer. That is exactly AC ≤ AB + BC. When the three points are not collinear, the detour is strictly longer, which is why a genuine (non-degenerate) triangle requires the strict inequality a + b > c for every pair of sides.

Interactive Graph

Formal Definition

Definition

For a triangle with side lengths a, b, c, the triangle inequality requires all three of the following simultaneously:

a + b > c

Sum of two sides exceeds the third

a + c > b

Same condition, second pairing

b + c > a

Same condition, third pairing

|a - b| < c < a + b

Equivalent single-line form: c is bounded above and below

Notation

Notation	Meaning
$a, b, c$	The three side lengths of a triangle (or three candidate positive lengths being tested)
$\|a-b\|$	Absolute difference of two side lengths; the third side must exceed this and be less than a+b

Properties

Triangle inequality (necessary and sufficient condition)

a + b > c, \quad a + c > b, \quad b + c > a

Condition: Three positive lengths a, b, c form a valid (non-degenerate) triangle if and only if all three inequalities hold; it suffices to check the largest side against the sum of the other two, since the other two inequalities hold automatically for positive lengths.

Example: Sides 5, 9, 12: 5+9=14>12 ✓, 5+12=17>9 ✓, 9+12=21>5 ✓ — valid triangle.

Degenerate case

\text{If } a+b = c \text{ exactly, the three points are collinear and the 'triangle' collapses into a straight segment.}

Range of a missing side

|a-b| < c < a+b

Condition: Given two sides a and b of a triangle, the third side c must lie strictly between their difference and their sum.

Applications

The triangle inequality is the defining axiom of a metric space, so it must hold for any valid distance function used in nearest-neighbor search, clustering, and pathfinding algorithms.

Worked Examples

Check the largest side (16) against the sum of the other two.
$7 + 10 = 17$
Since 17 > 16, the inequality holds; the other two pairings involving the largest side as an addend hold automatically for positive lengths.
$17 > 16$

Answer: Yes — 7 + 10 = 17 > 16, so a valid triangle exists.

Practice Problems

Difficulty 2/10

Can sides 3, 4, and 9 form a triangle?

Difficulty 3/10

A triangle has sides 8 and 15. Find the range of possible integer values for the third side x.

Difficulty 5/10

A surveyor measures two sides of a triangular plot as 40 m and 65 m. A colleague reports the third side is 110 m. Is this measurement possible?

Common Mistakes

Common Mistake

Checking only one pairing of sides (e.g. only a+b>c) and assuming that's sufficient.

Technically all three pairings must hold, though in practice it suffices to check the largest side against the sum of the other two — if that holds, the other two inequalities are automatic since the third side is even larger relative to a smaller sum.

Common Mistake

Using a non-strict inequality (≥) and thinking equality still gives a real triangle.

If a + b = c exactly, the three points are collinear and there is no triangle at all — only a degenerate, flattened case. The inequality must be strict for a genuine triangle.

Quiz

Summary

For any triangle with sides a, b, c: a+b>c, a+c>b, and b+c>a must all hold.
It suffices in practice to check the largest side against the sum of the other two.
Equality (a+b=c) gives a degenerate, collinear 'triangle' — not a genuine one.
Given two sides a and b, the third side c must satisfy |a-b| < c < a+b.
The triangle inequality generalizes to the defining axiom of any metric (distance function) in mathematics.

References

WebsiteWikipedia — Triangle inequality