Triangle Inequality Theorem

Rule explained

Can any 3 side lengths form a triangle?

No!It's actually not possible!

As you can see in the picture below, it's not possible to create a triangle that has side lengths of 4, 8, and 3

It turns out that there are some rules about the side lengths of triangles. You can't just make up 3 random numbers and have a triangle! You could end up with 3 lines like those pictured above that cannot be connected to form a triangle.

The Formula

The Triangle Inequality Theorem states that the sum of any 2 sides of a triangle must be greater than the measure of the third side.

Note: This rule must be satisfied for all 3 conditions of the sides.

In other words, as soon as you know that the sum of 2 sides is less than (or equal to) the measure of a third side, then you know that the sides do not make up a triangle.

You can experiment for yourself using our free online triangle inequality theorem calculator -- which lets you enter any three sides and explains how the triangle inequality theorem applies to them.

Do I have to always check all 3 sets?

NOPE!

You only need to see if the two smaller sides are greater than the largest side!

Look at the example above, the problem was that 4 + 3 (sum of smaller sides) is not greater than 10 (larger side)

We start using this shortcut with practice problem 2 below.

Interactive Demonstrations of Theorem

The interactive demonstration below shows that the sum of the lengths of any 2 sides of a triangle must exceed the length of the third side. The demonstration also illustrates what happens when the sum of 1 pair of sides equals the length of the third side--you end up with a straight line! You can't make a triangle!

Otherwise, you cannot create a triangle from the 3 sides.

A + B > C
6 + 6 > 6

A + C > B
6 + 6 > 6

B + C > A
6 + 6 > 6

Mouseover To Start Demonstration

Practice Problems

Problem 1
No

Use the triangle inequality theorem and examine all 3 combinations of the sides. As soon as the sum of any 2 sides is less than the third side then the triangle's sides do not satisfy the theorem.

Problem 2
Yes

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 5 + 6 > 7
Problem 3
No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 1.2 + 1.6 $$\color{Red}{ \ngtr }$$ 3.1
Problem 4
No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 6 + 8 $$\color{Red}{ \ngtr }$$ 16

More like Problem 1-4...

Problem 4.1
No

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 5 + 5 $$\color{Red}{ \ngtr }$$ 10
Problem 4.2
Yes

Use the shortcut and check if the sum of the 2 smaller sides is greater than the largest side.

 small + small > large because 7 + 9 > 15

Practice Problems Harder

Problem 5
You can use a simple formula shown below to solve these types of problems:

difference $$< x <$$ sum
$$8 -4 < x < 8+4$$

Answer: $$4 < x < 12$$

There's an infinite number of possible triangles, but we know that the side must be larger than 4 and smaller than 12 .

One Possible Solution

Here's an example of a triangle whose unknown side is just a little larger than 4:
Another Possible Solution

Here's an example of a triangle whose unknown side is just a little smaller than 12:
Problem 6

difference $$< x <$$ sum
$$7 -2 < x < 7+2$$

Answer: $$5 < x < 9$$
Problem 7

difference $$< x <$$ sum
$$12 -5 < x < 12 + 5$$

Answer: $$7 < x < 17$$