Triangle Inequality
Any two sides of a triangle must sum to more than the third.
About this topic
The triangle inequality says the sum of any two sides of a triangle must be greater than the third side. So sides of 3, 4 and 8 cannot form a triangle because 3 + 4 less than 8. If the sum equals the third side, the shape collapses into a straight line.
Given two sides a and b, the third side must be between the absolute difference of a and b and a + b. The principle says that a straight path is the shortest between two points, which is the basis of the idea of distance.
Worked example: can sides 5, 7 and 11 form a triangle? Check 5 + 7 = 12, which is greater than 11, so yes. For 4, 6 and 10, 4 + 6 = 10 is not greater, so the sides lie flat. If two sides are 5 and 9, the third side x must satisfy 9 − 5 less than x less than 9 + 5, so x is between 4 and 14. A stick of 14 would make a degenerate flat shape.
This is why the straight line is the shortest path between two points: going via a third point is never shorter. GPS routing and robot path planning use the same idea. A common error is checking only one pair of sides; you must be sure the two shortest sides together exceed the longest, which covers all three conditions. Remember it is strict inequality, so equality does not count as a triangle.
Key ideas
- a + b > c for every pair
- Used to test if a triangle can exist
- Largest side is opposite the largest angle
Quick check
Use cases: where Triangle Inequality helps
- Homework: review the key idea behind Triangle Inequality before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Triangle Inequality without notes.
- Visual learning: play with the interactive lab to see how Triangle Inequality changes when you adjust its inputs.
- Teaching a friend: explain Triangle Inequality in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Triangle Inequality?
Any two sides of a triangle must sum to more than the third.
What will I learn in Triangle Inequality?
a + b > c for every pair. Used to test if a triangle can exist. Largest side is opposite the largest angle.
Is Triangle Inequality part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Triangle Inequality lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Triangle Inequality?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.