Triangle Inequality Theorem Calculator

Understanding the Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental principle in geometry that determines whether three lengths can form a valid triangle. It states that the sum of any two sides must be greater than the third side.

The Three Conditions

For sides a, b, and c, all three must be true:

a + b > c (the two shorter sides together exceed the longest)
a + c > b
b + c > a

Note: If any sum equals the third side (not greater), the "triangle" degenerates into a straight line.

Finding the Third Side Range

Given two sides a and b, the third side c must satisfy:

|a - b| < c < a + b

The difference is the minimum, the sum is the maximum

Why It Works

Think of it physically: to connect three points with sticks of lengths a, b, and c, the two shorter sticks placed end-to-end must reach further than the longest stick. Otherwise, there's a gap and no triangle can form.

Frequently Asked Questions

What if two sides equal the third?

If a + b = c exactly, the three segments form a straight line (degenerate triangle) with zero area. This is not considered a valid triangle.

Why do I need to check all three conditions?

Actually, you only need to check that the sum of the two smaller sides exceeds the largest side. But checking all three is safer and ensures no calculation errors.

Does this work for any triangle?

Yes! The Triangle Inequality Theorem applies to all triangles - scalene, isosceles, equilateral, acute, right, and obtuse.

How is this related to the shortest path?

In geometry, "the shortest distance between two points is a straight line." Going via a third point is always longer, hence a + b > c.

Can any three positive numbers form a triangle?

No! For example, 1, 2, and 5 cannot form a triangle because 1 + 2 = 3 < 5. The numbers must satisfy the inequality.