Triangle inequality

Math

Definition

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. This rule determines whether three given lengths can form a valid triangle.

Examples

  • Sides of 3, 4, and 5 form a triangle because 3 + 4 > 5, 3 + 5 > 4, and 4 + 5 > 3
  • Sides of 1, 2, and 10 cannot form a triangle because 1 + 2 = 3 < 10
  • Determining the possible range for the third side of a triangle when two sides are 6 and 9: the third side must be between 3 and 15
Key Fact

For any triangle with sides a, b, c: a + b > c, a + c > b, and b + c > a

Related Terms

Triangles Pythagorean theorem Triangle congruence postulates

Study This Concept

Practice triangle inequality with free review games in these units:

Geometry Unit 5: Triangle Relationships