Triangle Inequality Theorem

Rule explained

Can any 3 side lengths form a triangle?

NOPE!

As you can see in the picture below, it's not possible to create a triangle that has side lengths of 4 , 10 , and 3

triangle inequality theorem, impossible triangle

It turns out that there are some rules about the side lengths of triangles. You can's 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 Tutorial

on the Triangle Inequality 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 eample above, the problem was that 4 + 3 (sum of smaller sides) is not greater than 10 (larger side)

Demonstrations

illustrating the Triangle Inequality 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
Problem 3

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
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
Problem 4

Do the side lengths below satisfy the triangle inequality theorem (ie can you make a triangle from the side lengths below?)

  • 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
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Problem 5

Do the side lengths below satisfy the triangle inequality theorem

  • 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

Practice Problems II

Problem 6

Two sides of a riangle 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 riangle 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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