Triangle Inequality Calculator

Test whether three side lengths can form a valid triangle using the Triangle Inequality Theorem.

Enter Three Side Lengths

Result

Triangle Validity
--
a + b > c --
a + c > b --
b + c > a --

Step-by-Step Analysis

The Triangle Inequality Theorem

The Triangle Inequality Theorem states that for any three lengths to form a valid triangle, the sum of any two sides must be strictly greater than the third side. This must hold for all three combinations of sides.

The Three Conditions

Condition 1

The sum of sides a and b must be greater than side c.

a + b > c

Condition 2

The sum of sides a and c must be greater than side b.

a + c > b

Condition 3

The sum of sides b and c must be greater than side a.

b + c > a

Special Cases

  • Degenerate triangle: If a + b = c (exactly equal), the three points are collinear and form a degenerate triangle with zero area.
  • All sides equal: An equilateral triangle always satisfies the inequality since a + a > a.
  • Very long side: If one side is equal to or greater than the sum of the other two, no triangle can be formed.
  • All side lengths must be positive numbers.

Practical Applications

The triangle inequality is used in computer science (metric spaces, graph algorithms), physics (vector addition), and engineering (structural stability analysis). It also applies to distances in any metric space.