The triangle inequality theorem says that in any triangle, the sum of the lengths of any two sides is greater than the length of the third side. So 3, 4 and 5 can form a triangle, but 2, 3 and 6 cannot, because 2 + 3 < 6. Shortcut: the two shorter sides must add to more than the longest.
In Spanish
teorema de la desigualdad del triángulo
Also called
triangle inequality, triangle inequality rule, third side rule
Grade level
Geometry
Grade 7GeometryCCSS 7.G.A.2SATACTGEDPSATHiSET
Triangle Inequality Theorem in a picture
Only when the two shorter sides add to more than the longest side can they lift off the base and meet in a point.
Formal definition and notation
For a triangle with sidelengthsa, b and c: a + b > c, a + c > b and b + c > a. Conversely, three positive numbers satisfying all three inequalities are the side lengths of some triangle. Equality, as in 3 + 4 = 7, gives three points on one line, a degenerate triangle with no area.
The Pythagorean theorem is an equation, a2 + b2 = c2, for right triangles only. The triangle inequality is an inequality, a + b > c, that holds for every triangle.
Triangle Inequality Theoremvs.Triangle angle sum
The angle sum theorem says the three angles add to 180°. The triangle inequality is about side lengths, not angles.
Quick check: 3 questions
Question 1 of 3
Which set of lengths can form a triangle?
Show answer and explanation
Answer: D.6 + 8 = 14 > 11. The others fail: 2 + 2 = 4 < 5, 3 + 6 = 9 is only equal to 9, and 1 + 10 = 11 < 12.
Question 2 of 3
Two sides of a triangle are 5 and 12. Which inequality describes the third side x?
Show answer and explanation
Answer: B. The third side is more than 12 − 5 = 7 and less than 12 + 5 = 17. The ≤ version allows the flat cases 7 and 17, and 0 < x forgets the lower limit.
Question 3 of 3
A triangle has sides 4 and 9, and the third side is a whole number. What is the largest it can be?
Show answer and explanation
Answer: B. The third side must be less than 4 + 9 = 13, so the largest whole number is 12. 13 would make the triangle flat, 14 is too long, and 5 equals the difference 9 − 4, which is too short.
What is the triangle inequality theorem in simple words?
Any two sides of a triangle, added together, must be longer than the third side. Picture three sticks: if the two short sticks together are shorter than the long stick, their ends cannot meet. In practice, add the two shortest lengths and check that the total is greater than the longest length.
How do you find the range of the third side?
Subtract the two known sides and add them. The third side must be greater than the difference and less than the sum. With sides 7 and 10, the third side is between 3 and 17, so 3 < x < 17. Neither endpoint is allowed, because those lengths make a flat figure.
Does the triangle inequality work for all triangles?
Yes. It holds for every triangle: acute, right, obtuse, scalene, isosceles and equilateral. For a right triangle it fits with the Pythagorean theorem: with legs 3 and 4, the hypotenuse 5 is less than 3 + 4 = 7. Even an equilateral triangle with sides 1 passes: 1 + 1 > 1.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.G.A.2
Cite this entry
Effortless Math Team. “Triangle Inequality Theorem.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/triangle-inequality-theorem/
Flashcard
TermTriangle Inequality Theorema + b > c
DefinitionAny two sides of a triangle add to more than the third. 2, 3, 6 fails because 2 + 3 < 6.
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