Triangle Inequality Theorem Calculator
Enter three side lengths to instantly check whether they satisfy the triangle inequality theorem. The calculator tests all three conditions, shows which ones pass and which fail, and explains by how much each condition is met or violated. You also get the valid numeric range for any one side given the other two, plus a step-by-step breakdown of the math.
What is the triangle inequality theorem?
The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side. Given sides a, b, and c, all three of the following must hold: a + b > c, a + c > b, and b + c > a. Equivalently, the longest side must be shorter than the combined length of the other two. If even one condition fails, the segments cannot be arranged to close into a triangle - they either overshoot or fall short of meeting at the third vertex.
How to check whether three sides form a triangle
Enter the three side lengths in any unit (the theorem is purely about ratios). The calculator evaluates each of the three inequalities and reports whether it passes or fails, together with the margin: how far the sum exceeds the third side (positive margin = passes) or how far it falls short (negative margin = fails). Only one failing condition is enough to rule out a triangle. The most practical shortcut is to check only the tightest condition: sort the three lengths so the longest is last, then confirm that the sum of the two shorter sides exceeds the longest. This single check is sufficient because the other two inequalities are automatically satisfied whenever the longest side is the bottleneck.
Finding the valid range for a missing side
If you know two sides and want to find what values the third can take, the theorem gives a strict two-sided bound. For sides a and b, the third side c must satisfy |a - b| < c < a + b. The lower bound prevents c from being so short that sides a and b would already overlap before c is inserted; the upper bound prevents c from being so long that a and b cannot stretch to meet it. Both endpoints are excluded - at exactly |a - b| or a + b the three points are collinear and the triangle degenerates to a line segment. This reverse-solve feature is built into the calculator: once you enter a and b it always shows the valid range for c.
Degenerate triangles and the strict inequality
A degenerate triangle occurs when one of the three inequalities is exactly at the boundary - that is, when the sum of two sides equals the third. The three vertices are then collinear (all on the same straight line), and the area is zero. Most geometry problems exclude this case by requiring the strict inequality (greater than, not greater than or equal). When the calculator shows a margin of exactly zero, the sides technically "pass" the arithmetic test but produce a flat, degenerate shape. The calculator flags this scenario so you can decide whether it is acceptable for your application.
Triangle inequality theorem - quick reference
| Condition | Must be | If violated |
|---|---|---|
| a + b > c | Strictly greater than c | The side c is too long; the other two cannot close the gap |
| a + c > b | Strictly greater than b | The side b is too long; the other two cannot close the gap |
| b + c > a | Strictly greater than a | The side a is too long; the other two cannot close the gap |
| a, b, c > 0 | All positive | Zero or negative sides have no geometric meaning |
| |a - b| < c < a + b | Valid range for c given a, b | If c equals |a-b| or a+b the triangle degenerates |
The three conditions that must ALL hold for sides a, b, c to form a valid (non-degenerate) triangle.
Frequently asked questions
Why do all three conditions need to be checked?
In practice only the tightest condition (involving the longest side) can fail, but checking all three is the rigorous approach and guards against input errors. If the three sides are sorted so that c is the largest, then a + b > c is the binding constraint; the other two hold automatically. If you do not know which side is longest, test all three.
Can a triangle have sides 3, 4, and 7?
No. Check the binding condition: 3 + 4 = 7, which is not strictly greater than 7 (it equals 7). The three points would be collinear, not a triangle. Any third side for the two-unit pair 3 and 4 must satisfy 1 < c < 7, so c = 7 is excluded.
What is the valid range for the third side if two sides are 5 and 8?
The third side c must satisfy |5 - 8| < c < 5 + 8, which simplifies to 3 < c < 13. Any value strictly between 3 and 13 (not including 3 or 13) will produce a valid triangle with the other two sides.
Does the triangle inequality theorem apply to right triangles?
Yes. The Pythagorean theorem tells you the exact relationship between the sides of a right triangle (a^2 + b^2 = c^2), but the triangle inequality theorem applies to all triangles including right triangles. Any set of sides that satisfies the Pythagorean equation automatically satisfies all three inequality conditions.
What is the reverse triangle inequality?
The reverse (or inverse) triangle inequality states that the absolute value of the difference of any two side lengths is less than the third side: |a - b| < c. This is the lower-bound half of the valid range formula and follows directly from rearranging the standard triangle inequality conditions. It is especially useful in proofs involving distances in metric spaces or absolute values.
Can the triangle inequality theorem be used with vectors?
Yes. In vector form, the theorem states that the magnitude of the sum of two vectors is at most the sum of their magnitudes: |u + v| <= |u| + |v|. This is a fundamental result in linear algebra and analysis. It extends to complex numbers, function spaces (via the Minkowski inequality), and many other mathematical contexts.