Triangle Inequality Theorem
Why some measurements cannot form a triangle.
By Geometry Editorial Team | 2026-06-02
Quick Answer
Triangle inequality requires every pair of sides to sum to more than the remaining side.
a + b > c, a + c > b, b + c > a
Table of Contents
Introduction
Use the Classifying Triangles Calculator homepage to test side sets and instantly see when inequality fails.
This theorem is the first quality gate for side-based classification and prevents invalid geometry conclusions.
Main Content
What is it?
Triangle inequality confirms that three lengths can physically connect to form a closed triangle.
It supports side classification in Triangle Classification by Sides and step-based workflows in How to Classify a Triangle.
Formula
Evaluate all three inequalities. Passing only one or two is not enough.
Step-by-step guide
- Order side lengths from smallest to largest.
- Check each pair sum against the third side.
- Reject the set if any inequality fails.
- Proceed to classification only after full validation.
Example
Sides 2, 3, 6 fail because 2 + 3 is not greater than 6, so no triangle exists.
Sides 4, 5, 8 pass all checks and can be classified.
FAQ
Why do worksheets include invalid side sets?
To verify you apply validation before classification.
Does inequality apply to angle-only input?
No, it is specifically for side-length validation.
Conclusion
Triangle inequality is a mandatory validation step, not an optional check.
Validate side sets now