What is the triangle inequality theorem?

Overview

Any polygonFigure with finite number of sides closed with three sides is known as a triangle. Triangle comes with many different angles and lengths. But how do we determine whether these three lengths will make a triangle? We can use the triangle inequality theorem to solve this.

The triangle inequality theorem states:

"Sum of two sides of a triangle is always greater than the third side."

Triangle inequality theorem formula

Consider an ∆ABC having sides with lengths a, b, and c, respectively. So, the triangle inequality theorem for this triangle is:

• $a<b+c$
• $b < c + a$
• $c < a + b$

Note: The triangle inequality theorem is pertinent for all types of triangles, for example, isosceles, equilateral, obtuse, scalene, and many more.

Examples

Let's look at two different examples below:

Example 1

Let's consider ∆ABC to understand the working of this theorem. As we can see in the slides below, the ∆ABC satisfies all the conditions of a triangle.

Example 2

Let's take another example and check whether a triangle with length 2,4,8 satisfies the theorem. As shown in the image below, it does not satisfy all the conditions. Hence, it is not a triangle.

Applications

The Triangle inequality theorem is widely used for rough estimation in:

• Urban planning
• Surveying
• Transportation
• Architectural designs