Triangle Inequality Calculator
Use the triangle inequality theorem to check three side lengths or find the possible range for a missing third side. The calculator explains the strict inequality behind the result.
What is the triangle inequality theorem?
The triangle inequality says that the sum of any two side lengths of a triangle must be greater than the third side. In everyday terms, two short sides must be long enough to meet and close the triangle. OpenStax discusses side and angle relationships in its overview of triangle properties and classifications.
This calculator handles both common homework versions: checking whether three side lengths form a triangle, and finding the possible range for a missing third side when two sides are known.
Triangle inequality rule
The inequalities must be strict. If the sum of two sides equals the third side, the shape collapses into a straight line. That is not a triangle because it has no interior area.
Worked examples
Why students lose marks on this topic
The most common mistake is using ≤ instead of < or >. A third side equal to the sum of the other two sides does not work. Another common mistake is checking only one pair of sides. For a valid triangle, all three pair-sum conditions must be true.
Common questions
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The triangle inequality theorem says the sum of any two sides of a triangle must be greater than the third side. For sides a, b, and c, you must have a + b > c, a + c > b, and b + c > a. If even one of those statements is false, the side lengths cannot make a triangle.
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If two sides are a and b, the third side x must be greater than the difference of the two sides and less than their sum. The range is |a - b| < x < a + b. For example, if two sides are 5 and 9, the third side must be greater than 4 and less than 14.
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If the third side equals the sum of the other two sides, the three segments form a straight line, not a triangle. A triangle needs the sides to close with a positive area. That is why the inequality must be strict, using greater than rather than greater than or equal to.
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When checking three given sides, yes. In practice, if you sort the sides from smallest to largest, it is enough to check whether the two smaller sides add to more than the largest side. However, writing all three inequalities is often clearer for homework and helps show the reasoning.
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Yes. Side lengths do not have to be whole numbers. The triangle inequality works the same way for decimals and fractions. The only requirement is that all side lengths are positive and that the sum of the two shorter sides is greater than the longest side.
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No. It only tells whether the side lengths can form a triangle or what range a missing side can have. To find angles, you would need additional tools such as the law of cosines, trigonometry, or special triangle relationships.