Triangle Inequality Formula

Triangle inequality is the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

The Formula

a+b>c   for every pair of sidesa+b>c\;\text{ for every pair of sides}

When to use: Try to build a triangle with two short sticks and one very long one—you can't. The two short sticks can't reach across to close the shape. It's like trying to take a shortcut: the direct path (one side) is always shorter than going around (the other two sides combined).

Quick Example

Sides 3,4,53, 4, 5: valid because 3+4>53 + 4 > 5, 3+5>43 + 5 > 4, 4+5>34 + 5 > 3. Sides 1,2,51, 2, 5: invalid because 1+2=3≯5\text{invalid because } 1 + 2 = 3 \not> 5

Notation

The sum of any two side lengths must be greater than the third side.

What This Formula Means

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

Try to build a triangle with two short sticks and one very long one—you can't. The two short sticks can't reach across to close the shape. It's like trying to take a shortcut: the direct path (one side) is always shorter than going around (the other two sides combined).

Formal View

∀ a,b,c>0\forall\, a, b, c > 0 forming a triangle: ∣a−b∣<c<a+b|a - b| < c < a + b (and cyclic permutations); this is the triangle inequality for the Euclidean metric: d(P,R)≤d(P,Q)+d(Q,R)d(P,R) \leq d(P,Q) + d(Q,R), strict unless P,Q,RP, Q, R are collinear

Worked Examples

Example 1

easy
Can sides of length 4, 7, and 10 form a triangle? Check all three triangle inequality conditions.

Answer

Yes, sides 4, 7, and 10 can form a triangle.

First step

1
Step 1: The Triangle Inequality Theorem states that the sum of any two sides must be greater than the third side. Check all three combinations.

Full solution

  1. 2
    Step 2: 4+7=11>104 + 7 = 11 > 10 ✓
  2. 3
    Step 3: 4+10=14>74 + 10 = 14 > 7 ✓
  3. 4
    Step 4: 7+10=17>47 + 10 = 17 > 4 ✓ All three conditions are satisfied.
The Triangle Inequality requires all three pairwise sums to exceed the remaining side. In practice, only the sum of the two shortest sides needs to exceed the longest side — if that holds, the other two conditions are automatically satisfied. Here, 4+7=11>104 + 7 = 11 > 10 is the critical check.

Example 2

medium
Two sides of a triangle have lengths 8 and 13. Find the range of possible lengths for the third side.

Example 3

medium
Two sides of a triangle measure 99 and 1414. List all integer values the third side could take.

Common Mistakes

  • Checking only one random pair — compare the two shortest sides against the longest side.
  • Using greater than or equal — equality makes a flat line, not a triangle.
  • Confusing possible triangle with right triangle — possible does not mean right.

Why This Formula Matters

Triangle inequality prevents impossible geometry. It supports construction, proof, distance reasoning, and later comparisons in coordinate geometry. Recognizing it by "Is the sum of the two shorter sides greater than the longest side?" — rather than by familiar numbers — is what lets a student tell it apart from pythagorean theorem and triangle angle sum in a mixed problem set.

Frequently Asked Questions

What is the Triangle Inequality formula?

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

How do you use the Triangle Inequality formula?

Try to build a triangle with two short sticks and one very long one—you can't. The two short sticks can't reach across to close the shape. It's like trying to take a shortcut: the direct path (one side) is always shorter than going around (the other two sides combined).

What do the symbols mean in the Triangle Inequality formula?

The sum of any two side lengths must be greater than the third side.

Why is the Triangle Inequality formula important in Math?

Triangle inequality prevents impossible geometry. It supports construction, proof, distance reasoning, and later comparisons in coordinate geometry. Recognizing it by "Is the sum of the two shorter sides greater than the longest side?" — rather than by familiar numbers — is what lets a student tell it apart from pythagorean theorem and triangle angle sum in a mixed problem set.

What do students get wrong about Triangle Inequality?

The procedure for triangle inequality is the easy part; the trap is checking only one random pair. Asking "Is the sum of the two shorter sides greater than the longest side?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Triangle Inequality formula?

Before studying the Triangle Inequality formula, you should understand: triangles, addition, comparison.

← Back to Triangle Inequality See All Examples Practice Problems