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Triangle Inequality

Triangle Inequality

A mathematical restatement of the concept that the shortest distance between two points is a straight line. The triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

 

Triangle with vertices A, B, C and sides labeled a, b, c. Inequalities: a+b>c, a+c>b, b+c>a.

 

 

See also

Triangle inequality with absolute value

Key Formula

a+b>c,a+c>b,b+c>aa + b > c, \quad a + c > b, \quad b + c > a
Where:
  • aa = Length of the first side of the triangle
  • bb = Length of the second side of the triangle
  • cc = Length of the third side of the triangle

Worked Example

Problem: Can a triangle have sides of length 5, 8, and 12?
Step 1: Identify the three sides. Let a = 5, b = 8, and c = 12.
a=5,b=8,c=12a = 5, \quad b = 8, \quad c = 12
Step 2: Check the first condition: is a + b > c?
5+8=13>12✓5 + 8 = 13 > 12 \quad \checkmark
Step 3: Check the second condition: is a + c > b?
5+12=17>8✓5 + 12 = 17 > 8 \quad \checkmark
Step 4: Check the third condition: is b + c > a?
8+12=20>5✓8 + 12 = 20 > 5 \quad \checkmark
Step 5: All three inequalities hold, so these side lengths can form a triangle.
Answer: Yes, a triangle with sides 5, 8, and 12 is valid because all three triangle inequality conditions are satisfied.

Another Example

This example reverses the typical problem: instead of checking given sides, you use the triangle inequality to find all possible values for an unknown side.

Problem: Two sides of a triangle measure 6 and 10. Find the range of possible lengths for the third side.
Step 1: Let the known sides be a = 6 and b = 10, and let the unknown side be c.
a=6,b=10,c=  ?a = 6, \quad b = 10, \quad c = \;?
Step 2: Apply the inequality a + b > c to find the upper bound for c.
6+10>c  ⟹  c<166 + 10 > c \implies c < 16
Step 3: Apply the inequality a + c > b. Rearrange to find the lower bound for c.
6+c>10  ⟹  c>46 + c > 10 \implies c > 4
Step 4: The third inequality b + c > a gives c > −4, which is automatically true for any positive length, so it adds no new constraint.
10+c>6  ⟹  c>−4(always true)10 + c > 6 \implies c > -4 \quad \text{(always true)}
Step 5: Combine the bounds. The third side must be strictly between 4 and 16.
4<c<164 < c < 16
Answer: The third side must have a length greater than 4 and less than 16.

Frequently Asked Questions

What happens when the sum of two sides equals the third side?
If a + b = c exactly, the three points are collinear — they lie on a straight line — and no triangle is formed. The triangle inequality requires a strict inequality (greater than, not greater than or equal to). A segment of length 0 at the 'vertex' is not a valid triangle.
Do you need to check all three inequalities?
Technically all three must hold, but there is a shortcut. If you add the two shortest sides and show their sum exceeds the longest side, the other two inequalities are automatically satisfied. So in practice, you only need one check: the sum of the two smaller sides must be greater than the largest side.
What is the difference between the triangle inequality and the triangle inequality with absolute value?
The geometric triangle inequality deals with side lengths of a triangle. The absolute value version, |a + b| ≤ |a| + |b|, is an algebraic statement about real numbers (or vectors). They share the same underlying idea — a direct path is never longer than an indirect path — but they appear in different branches of math.

Triangle Inequality (Geometry) vs. Triangle Inequality with Absolute Value

Triangle Inequality (Geometry)Triangle Inequality with Absolute Value
DefinitionThe sum of any two sides of a triangle exceeds the third side.For any real numbers a and b, |a + b| ≤ |a| + |b|.
Formulaa + b > c (and cyclic permutations)|a + b| ≤ |a| + |b|
Typical useDetermining whether three lengths can form a triangle, or finding the range of a missing side.Proving inequalities in algebra, analysis, and proofs involving distances or norms.
Inequality directionStrict inequality (>); equality means a degenerate triangle.Non-strict inequality (≤); equality when a and b have the same sign.

Why It Matters

The triangle inequality appears in geometry courses whenever you construct or analyze triangles — for instance, determining if a set of measurements from a real-world structure can actually form a triangle. It is also foundational in more advanced mathematics: the concept extends to distances in coordinate geometry, vector spaces, and metric spaces, making it one of the most widely used inequalities across all of mathematics.

Common Mistakes

Mistake: Only checking one pair of sides instead of confirming the critical pair (the two shortest sides summed against the longest).
Correction: Always identify the longest side first. If the sum of the two shorter sides exceeds the longest, all three conditions are guaranteed. If you pick a different pair, you might pass a weaker check while missing the binding constraint.
Mistake: Using ≥ (greater than or equal to) instead of > (strictly greater than).
Correction: The triangle inequality requires a + b to be strictly greater than c. If a + b = c, the figure collapses into a line segment (a degenerate triangle), which is not a valid triangle.

Related Terms