Triangle Inequality Formula

The Formula

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

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

Notation

a, b, c are the three side lengths; > means strictly greater than

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

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

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.

Common Mistakes

  • Only checking one pair of sides instead of all three
  • Using \geq instead of > (equality gives a degenerate 'flat' triangle, not a real one)
  • Confusing with the Pythagorean theorem, which only applies to right triangles

Why This Formula Matters

A fundamental feasibility check in geometry and a key inequality that generalizes to distance in all of mathematics.

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?

a, b, c are the three side lengths; > means strictly greater than

Why is the Triangle Inequality formula important in Math?

A fundamental feasibility check in geometry and a key inequality that generalizes to distance in all of mathematics.

What do students get wrong about Triangle Inequality?

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

What should I learn before the Triangle Inequality formula?

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

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