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Triangle Inequality Theorem

Rule explained

Can any 3 side lengths form a triangle?

For instance, can I create a triangle from sides of length...say 4, 8 and 3?

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

triangle inequality theorem, impossible triangle

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.

Video On Theorem

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.

Triangle Inequality Theorem Picture and Formula
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.

Triangle Inequality Theorem Example 1

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 Denomstrations 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.

Demonstration 1

When the sum of 1 pair of sides exactly equals the measure of a 3rd side.

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Demonstration 2

When the sum of 1 pair of sides is less than the measure of a 3rd side.

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Practice Problems

Problem 1

Could a triangle have side lengths of

  • Side 1: 4
  • Side 2: 8
  • Side 3: 2
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.

triangle inequality diagram
Problem 2

Could a triangle have side lengths of

  • Side 1: 5
  • Side 2: 6
  • Side 3: 7
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 Green Check
Problem 3

Could a triangle have side lengths of

  • Side 1: 1.2
  • Side 2: 3.1
  • Side 3: 1.6
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 Red X
Problem 4

Could a triangle have side lengths of

  • Side 1: 6
  • Side 2: 8
  • Side 3: 15
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 Red X

More like Problem 1-4...

Problem 4.1

Could a triangle have side lengths of

  • Side 1: 5
  • Side 2: 5
  • Side 3: 10
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 Red X
Problem 4.2

Could a triangle have side lengths of

  • Side 1: 7
  • Side 2: 9
  • Side 3: 15
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 Green Check

Practice Problems Harder

Problem 5

Two sides of a triangle have lengths 8 and 4. Find all possible lengths of the third side.

Triangle

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:
Triangle
Another Possible Solution

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

Two sides of a triangle have lengths 2 and 7. Find all possible lengths of the third side.

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

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

Two sides of a triangle have lengths 12 and 5. Find all possible lengths of the third side.

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

Answer: $$7 < x < 17$$

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