Triangle Inequality Theorem Calculator: Check Valid Triangles (2026)
Check if three side lengths can form a valid triangle. Find the range of possible third side values with step-by-step verification.
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Put these formulas into practice with our instant, step-by-step Triangle Inequality Theorem Calculator.
The triangle inequality theorem is one of the most fundamental results in geometry: the sum of any two sides of a triangle must be greater than the third side. Our calculator verifies whether three given side lengths form a valid triangle, and when you know two sides, it finds the range of possible values for the third side. This theorem is essential for validating measurements, solving geometry problems, and understanding the limits of triangular shapes.
Key Takeaway
For three sides a, b, c to form a valid triangle: a + b > c, a + c > b, and b + c > a. All three inequalities must hold. Given sides a and b, the third side c must satisfy |a - b| < c < a + b.
The Triangle Inequality Theorem
The triangle inequality guarantees that three line segments can be connected to form a closed triangular shape. If any inequality fails, the segments cannot meet to form a triangle. This is not just a mathematical curiosity — it is a practical requirement for any construction, surveying, or engineering project involving triangular structures.
Finding the Range of the Third Side
When you know two sides a and b (where a ≤ b), the third side c must satisfy b - a < c < b + a. The minimum is |a - b| (exclusive) and the maximum is a + b (exclusive). For example, with sides 3 and 5, the third side must be between 2 and 8.
What Is a Degenerate Triangle?
A degenerate triangle has three collinear points (all on the same line) and zero area. It occurs when one side equals the sum of the other two (e.g., 3, 5, 8). The triangle inequality uses strict inequalities, so degenerate triangles are not considered valid.
Frequently Asked Questions
- Can the sum of two sides equal the third?
- No. If a + b = c, the points are collinear and form a degenerate triangle with zero area.
- How many inequalities must be checked?
- Three. But if you sort sides so a ≤ b ≤ c, only a + b > c needs checking.
- Is this related to the shortest path?
- Yes. The triangle inequality guarantees that a straight line is the shortest path between two points.
Conclusion
The triangle inequality theorem determines whether three lengths can form a valid triangle. Use our free triangle inequality theorem calculator to verify triangles and find valid side ranges.