Triangle Inequality Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Triangle Inequality.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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).

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The shortest path between two points is a straight line, so one side of a triangle (the direct path) must be shorter than the other two sides combined (the indirect path). This is why not every set of three lengths can form a triangle.

Common stuck point: You must check all three combinations, though in practice checking that the sum of the two shortest sides exceeds the longest is sufficient.

Worked Examples

Example 1

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

Solution

  1. 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.
  2. 2
    Step 2: 4 + 7 = 11 > 10 ✓
  3. 3
    Step 3: 4 + 10 = 14 > 7 ✓
  4. 4
    Step 4: 7 + 10 = 17 > 4 ✓ All three conditions are satisfied.

Answer

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

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which of these sets of side lengths cannot form a triangle? (a) 3, 4, 5. (b) 1, 2, 3. (c) 5, 8, 12.

Example 2

hard
In \triangle ABC, AB = 6 and BC = 10. The perimeter is 24. Is this a valid triangle? Find AC and verify.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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