Triangle Inequality
Any two sides sum to more than the third.
Key idea
Any two sides sum to more than the third.
In depth
In any triangle, the sum of two sides is greater than the third. A straight path is the shortest between two points, so a detour through the third vertex cannot be shorter. It tests whether three lengths can form a triangle.
Worked example
3, 4, 8: 3 + 4 = 7 < 8, so no triangle. 3, 4, 5: 3 + 4 > 5, works.
Formula
Real-world uses
- Checking whether three sticks can form a triangle
- Estimating the shortest walking route
Common mistake
Checking only one pair instead of confirming the largest side is less than the sum of the other two.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Triangle Inequality?
Any two sides sum to more than the third. It is a topic at the intermediate level in Geometry.
What is the formula for Triangle Inequality?
The key formula is a + b > c.
Where is Triangle Inequality used in real life?
Real-world examples: Checking whether three sticks can form a triangle; Estimating the shortest walking route.
What should I learn before Triangle Inequality?
Make sure you are comfortable with Exterior Angle Theorem, the previous topic in Geometry.
What is a common mistake with Triangle Inequality?
Checking only one pair instead of confirming the largest side is less than the sum of the other two.
How can I practise Triangle Inequality?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Triangle Inequality?
Move on to Congruent Triangles next.
Related intermediate topics in Geometry
- Vertical Angles – Opposite angles formed by intersecting lines are equal.
- Parallel Lines and Transversals – Corresponding and alternate interior angles are equal.
- Exterior Angle Theorem – An exterior angle equals the sum of the two remote interior angles.
- Congruent Triangles – Same shape and size: SSS, SAS, ASA, AAS, HL.
- Similar Triangles – Same shape, proportional sides: AA, SAS~, SSS~.
- Pythagorean Theorem – In a right triangle the squares on the legs sum to the square on the hypotenuse.