Triangle Inequality Theorem Calculator

Determine if three side lengths can form a valid triangle.

Enter the lengths of three sides (A, B, and C) to check if they satisfy the triangle inequality theorem: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Examples

Click on any example to load it into the calculator.

Valid Scalene Triangle

valid-scalene

A common case where all side lengths are different and form a valid triangle.

Side A: 5

Side B: 7

Side C: 10

Valid Isosceles Triangle

valid-isosceles

An example with two equal sides that form a valid triangle.

Side A: 8

Side B: 8

Side C: 12

Valid Equilateral Triangle

valid-equilateral

An example with all three sides equal.

Side A: 6

Side B: 6

Side C: 6

Invalid Triangle

invalid

A case where the sum of two sides is not greater than the third, failing the theorem.

Side A: 3

Side B: 4

Side C: 8

Other Titles
Understanding the Triangle Inequality Theorem: A Comprehensive Guide
Explore the fundamental principle governing triangle side lengths, its proofs, and its wide-ranging applications in mathematics and beyond.

What is the Triangle Inequality Theorem?

  • The sum of any two sides of a triangle must be greater than the third side.
  • A fundamental rule for determining if three lengths can form a triangle.
  • Geometrically, it means the shortest path between two points is a straight line.
The Triangle Inequality Theorem is a core concept in geometry that defines the relationship between the three side lengths of any triangle. It states that for a triangle with side lengths a, b, and c, the following three inequalities must hold true:
1. a + b > c
2. a + c > b
3. b + c > a
If even one of these conditions is not met, the three lengths cannot form a closed triangular shape. In essence, the theorem ensures that the sides can connect to form a triangle. It's a simple yet powerful rule that underpins many geometric proofs and applications.

Theorem Conditions

  • For sides 5, 7, 10: 5+7 > 10 (12 > 10), 5+10 > 7 (15 > 7), 7+10 > 5 (17 > 5). Valid.
  • For sides 3, 4, 8: 3+4 > 8 (7 > 8) is false. Invalid.

Step-by-Step Guide to Using the Triangle Inequality Theorem Calculator

  • Enter the three side lengths into the designated fields.
  • Click the 'Check Triangle' button to perform the validation.
  • Analyze the results to understand the verdict and classification.
Input Guidelines
1. Side A: Enter the length of the first side. This must be a positive number.
2. Side B: Enter the length of the second side. This must also be a positive number.
3. Side C: Finally, enter the length of the third side, which must be a positive number.
Interpreting the Results
  • Verdict: This will clearly state whether the provided lengths can form a valid triangle.
  • Reason: If the triangle is invalid, this section will specify which of the three inequalities was violated.
  • Triangle Type: If the triangle is valid, the calculator will classify it as Equilateral (all sides equal), Isosceles (two sides equal), or Scalene (all sides different).

Practical Examples

  • Input: A=8, B=15, C=17 -> Output: Valid, Scalene (This is also a right triangle!)
  • Input: A=10, B=10, C=10 -> Output: Valid, Equilateral
  • Input: A=5, B=5, C=10 -> Output: Invalid (Reason: 5+5 > 10 is false)

Real-World Applications of the Triangle Inequality Theorem

  • Essential in GPS navigation and distance calculation.
  • Used in engineering for structural stability analysis.
  • A key principle in network routing algorithms.
Navigation and GPS
In GPS systems, the theorem helps in calculating the shortest distance between two points on the Earth's surface. The direct path (a straight line) is always shorter than any indirect path that involves a third point, forming a triangle.
Engineering and Architecture
Engineers use the theorem to ensure the stability of structures like bridges and trusses. A triangular frame is inherently stable, and the theorem guarantees that the sum of the lengths of any two structural members must exceed the third to form a rigid shape.
Computer Science and Networking
In network routing, the 'cost' or 'latency' between nodes can be thought of as side lengths. The triangle inequality ensures that the direct path from node A to node C is always faster or cheaper than routing through an intermediate node B.

Industry Use Cases

  • A flight path from New York to London is shorter than flying from New York to Iceland and then to London.
  • In robotics, a robot's path planning algorithm uses the theorem to find the most efficient route.
  • Telecommunication networks use it to optimize data packet routing.

Common Misconceptions and Correct Methods

  • Checking only one inequality is not enough.
  • Side lengths can be decimals, not just integers.
  • A 'degenerate' triangle (a + b = c) is a straight line, not a true triangle.
Misconception 1: Checking Only One Condition
A common mistake is to only check if the sum of the two shorter sides is greater than the longest side. While this is a useful shortcut, the formal definition requires checking all three inequalities to be mathematically rigorous. Our calculator checks all three for correctness.
Misconception 2: What about a + b = c?
If the sum of two sides equals the third (e.g., sides 3, 4, 7), they form what is called a 'degenerate triangle'. This is simply a straight line segment, as the two shorter sides lie perfectly flat against the longest one. It does not have an interior area and is not considered a true triangle.
Misconception 3: Zero or Negative Lengths
By definition, the length of a side of a triangle must be a positive value. Zero or negative lengths are not physically possible for a triangle's side, and our calculator will validate this.

Clarifying Concepts

  • Sides 7, 3, 5: You must check 7+3>5, 7+5>3, AND 3+5>7. All are true.
  • Sides 2, 8, 4: Just checking 2+4>8 is enough to see it's invalid. But rigorously, you would check all three.
  • Sides 5, 12, 13: A valid right triangle, satisfying all three conditions.

Mathematical Derivation and Proofs

  • Geometric proof using a compass and straightedge.
  • Vector proof based on the properties of vector norms.
  • The theorem is a property of metric spaces in advanced mathematics.
Geometric Proof (Euclidean)
The most intuitive proof comes from Euclid's Elements. The axiom that 'a straight line is the shortest distance between two points' is the foundation. Given a triangle ABC, the straight path from A to C (side AC) must be shorter than the path that goes from A to B and then to C (sides AB + BC). Therefore, AB + BC > AC. This logic can be applied to all three side combinations.
Vector Proof
In vector algebra, if we represent the sides of a triangle as vectors such that u + v + w = 0, we can define the side lengths as the magnitudes (or norms) of these vectors: a = ||u||, b = ||v||, c = ||w||. Since u + v = -w, we have ||u + v|| = ||-w|| = ||w|| = c. A fundamental property of vector norms is that ||u + v|| ≤ ||u|| + ||v||. Therefore, c ≤ a + b. The equality holds only when the vectors are collinear (forming a degenerate triangle), so for any non-degenerate triangle, c < a + b.

Proof Illustrations

  • Try drawing a triangle with sides 10cm, 5cm, and 4cm. The two shorter sides won't be able to meet.
  • Consider vectors u=(3,0) and v=(0,4). Then u+v=(3,4). ||u||=3, ||v||=4, ||u+v||=5. We see 3+4 > 5.