Triangle Inequality

Geometry
principle

Also known as: triangle inequality theorem, side length rule

Grade 6-8

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The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. A fundamental feasibility check in geometry and a key inequality that generalizes to distance in all of mathematics.

Definition

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

💡 Intuition

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).

🎯 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.

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

Formula

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

Notation

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

🌟 Why It Matters

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

💭 Hint When Stuck

Sort the three side lengths from smallest to largest. Then check if the sum of the two smallest sides is greater than the largest side. If yes, the triangle is valid. If not, no triangle can be formed.

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

🚧 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.

⚠️ 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

Frequently Asked Questions

What is Triangle Inequality in Math?

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

What is the Triangle Inequality formula?

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

When do you use Triangle Inequality?

Sort the three side lengths from smallest to largest. Then check if the sum of the two smallest sides is greater than the largest side. If yes, the triangle is valid. If not, no triangle can be formed.

How Triangle Inequality Connects to Other Ideas

To understand triangle inequality, you should first be comfortable with triangles, addition and comparison. Once you have a solid grasp of triangle inequality, you can move on to pythagorean theorem.

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