Triangle Inequality Theorem Calculator

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. All three inequalities must hold: a + b > c, a + c > b, and b + c > a.

It determines whether three lengths can form a valid triangle. Without it, three segments might not connect to form a closed shape. It is fundamental to geometry, vector mathematics, and distance metrics.

Given 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 sides 3 and 5: 2 < c < 8.

No. If a + b = c, the three points are collinear (on the same line) and form a degenerate triangle with zero area. The inequality is strict: a + b > c, not a + b ≥ c.

A degenerate triangle has three collinear points (all on the same line). Its area is zero. It occurs when one side equals the sum of the other two. It is not considered a valid triangle in most contexts.

Subtract: lower bound = |a - b|, upper bound = a + b. The third side must be strictly between these values. For sides 7 and 10: 3 < third side < 17.

Yes. The triangle inequality applies to all triangles — scalene, isosceles, equilateral, right, acute, and obtuse. Every valid triangle must satisfy all three inequalities.

For a right triangle, a² + b² = c². The triangle inequality says a + b > c. These are different conditions: the Pythagorean theorem specifies the exact relationship for right triangles, while the triangle inequality is a general requirement for all triangles.

Yes. Surveyors, architects, and engineers use the triangle inequality to verify that measured distances can form valid triangles. If measurements violate the inequality, there is likely a measurement error.

If any inequality fails (e.g., a + b ≤ c), the three side lengths cannot form a valid triangle. The segments would not connect to form a closed shape. The calculator will report this as an invalid triangle.

The triangle inequality for vectors states |a⃗ + b⃗| ≤ |a⃗| + |b⃗|. The magnitude of a sum is at most the sum of magnitudes. Equality holds only when the vectors point in the same direction.

For n points, the distance between any two must be less than or equal to the sum of distances through intermediate points: d(A,C) ≤ d(A,B) + d(B,C). This is the foundation of metric spaces.

Three inequalities for a triangle: a + b > c, a + c > b, b + c > a. However, if you sort the sides so a ≤ b ≤ c, you only need to check a + b > c (the other two are automatically satisfied).

The triangle inequality guarantees that a straight line is the shortest path between two points. Any detour through a third point is at least as long: d(A,C) ≤ d(A,B) + d(B,C).

Yes. Beyond just checking validity, the relationship between a² + b² and c² classifies the triangle: a² + b² > c² (acute), a² + b² = c² (right), a² + b² < c² (obtuse). This supplements the triangle inequality.

Triangle inequality is used to validate mesh triangles in 3D modeling. Invalid triangles (from import errors or floating-point issues) can cause rendering artifacts. Graphics engines check this during mesh processing.

Among all triangles with a given perimeter, the equilateral triangle has the largest area. The inequality states: A ≤ (P²√3)/36, where P is the perimeter and A is the area. Equality holds for equilateral triangles.

The Law of Cosines: c² = a² + b² - 2ab cos(C). Since cos(C) ranges from -1 to 1, we get (a-b)² ≤ c² ≤ (a+b)², which is equivalent to |a-b| < c < a+b — the triangle inequality.

If one side is very small (approaching 0), the triangle approaches a degenerate state. The triangle inequality still requires the small side to be greater than |a - b|. For sides 5, 7, and 0.1: 5 + 0.1 = 5.1 < 7, so this is INVALID.

Yes! If a = b = c, all three inequalities become a + a > a — 2a > a, which is true for any positive a. This is the equilateral triangle — the most symmetric valid triangle.

The Triangle Inequality Theorem Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Triangle Inequality Theorem Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.