Triangle Inequality Theorem Calculator
Instantly check whether three side lengths can form a valid triangle and identify the triangle type.
Enter the three side lengths and click Check. The calculator verifies all three triangle inequality conditions and classifies the result.
Triangle Inequality Theorem Calculator
Instantly check whether three side lengths can form a valid triangle and identify the triangle type.
About the Triangle Inequality Theorem Calculator
The triangle inequality theorem is one of the most fundamental rules in Euclidean geometry. It states that for any three lengths a, b, and c to form a triangle, each of the following three conditions must hold simultaneously: a + b > c, a + c > b, and b + c > a. If even one condition fails, the three segments cannot be arranged to form a closed triangular shape — the ends simply do not meet.
The intuition behind the theorem is straightforward: the shortest path between two points is a straight line. Suppose you want to travel from vertex A to vertex C. You can go directly along side b (the segment AC), or you can take a detour through vertex B by following sides a and c. The direct path b must always be shorter than the detour a + c; if it were not, the triangle would be degenerate or impossible. The same logic applies to the other two side combinations.
A special case arises when a + b = c (or any permutation). The three points are collinear — they lie on a straight line — and the "triangle" degenerates to a line segment with zero area. This is called a degenerate triangle. It technically satisfies the theorem with a weak inequality (≤), but most geometric contexts require a strict inequality (>) to obtain a proper triangle with positive area.
Once the theorem confirms the sides are valid, the triangle can be classified by its side lengths. If all three sides are equal (a = b = c), the triangle is equilateral. If exactly two sides are equal, it is isosceles. If all three sides are different, it is scalene. The calculator displays this classification alongside the validity verdict.
The triangle inequality theorem has applications far beyond textbook geometry. In GPS and navigation systems, it underpins the shortest-path principle: the direct route from A to C is always shorter than any route through an intermediate point B. Network engineers use it to reason about routing latency. Robot path-planning algorithms rely on it to prune impossible configurations. In structural engineering, the stability of a triangular truss depends on all three members satisfying the theorem. Understanding when and why three lengths cannot form a triangle is essential knowledge for anyone working in geometry, physics, engineering, or computer science.
Triangle Inequality Examples
Examples showing both valid and invalid side combinations with explanations.
| Sides (a, b, c) | Result | Explanation |
|---|---|---|
| a = 5, b = 7, c = 10 | Valid — Scalene | 5+7=12>10, 5+10=15>7, 7+10=17>5. All three conditions satisfied; all sides differ so it is scalene. |
| a = 10, b = 10, c = 10 | Valid — Equilateral | 10+10=20>10 for every pair. All sides equal, so equilateral. |
| a = 3, b = 4, c = 8 | Invalid | 3+4=7, which is NOT greater than 8. The two shorter sides cannot reach across the longest side. |
| a = 8, b = 15, c = 17 | Valid — Scalene | 8+15=23>17, 8+17=25>15, 15+17=32>8. This is also a right triangle (8²+15²=17²). |
How to Use the Triangle Inequality Theorem Calculator
- Enter positive numbers for all three side lengths (Side A, Side B, Side C) in the input fields.
- Click "Check Triangle". The calculator verifies all three inequality conditions: a+b > c, a+c > b, and b+c > a.
- Read the verdict: "Valid Triangle" means a triangle can be formed; "Not a Valid Triangle" means at least one condition failed.
- If valid, the calculator also shows the triangle type: equilateral, isosceles, or scalene based on which sides are equal.
- If invalid, the reason message identifies which specific inequality was violated so you can understand why the sides cannot form a triangle.
Triangle Inequality Theorem FAQ
What exactly is the triangle inequality theorem?
The theorem states that for three positive lengths a, b, c to form a triangle, the sum of any two sides must be strictly greater than the third side. All three conditions (a+b > c, a+c > b, b+c > a) must hold. If any one fails, the three lengths cannot close into a triangle.
Do I need to check all three conditions or just one?
Formally, all three must be checked. In practice, if you know which side is longest, it is sufficient to verify that the sum of the two shorter sides exceeds it, because the other two conditions will then be automatically satisfied. The calculator checks all three for mathematical rigour.
What happens when a + b equals c exactly?
When a + b = c, the three points are collinear and the shape degenerates into a straight line segment. There is no enclosed area, and it is not considered a proper triangle. A valid triangle requires the strict inequality a + b > c.
Can side lengths be decimals or fractions?
Yes. The triangle inequality applies to any positive real numbers, not just integers. Enter decimal values such as 2.5, 7.8, or 0.1 just as you would integers. The calculator evaluates the inequalities accurately for any valid numeric input.
How is the triangle type determined after validation?
After confirming the sides form a valid triangle, the calculator compares the three lengths. If all three are equal, it is equilateral. If exactly two are equal, it is isosceles. If all three are different, it is scalene. Note that a right triangle can be scalene or isosceles depending on its angles.
Why does the triangle inequality matter in engineering?
In structural engineering, a triangular truss is inherently rigid and stable. The triangle inequality ensures that the three members can actually be assembled into a closed frame. In network routing and GPS, the theorem guarantees that a direct path is never longer than an indirect path through a relay point, which is the basis of shortest-path algorithms.