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
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.
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 usingour 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
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.
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: 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 |
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 |
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 |
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 |
Practice Problems Harder
Problem 5
Two sides of a triangle have lengths 8 and 4. Find all possible lengths of the third side.
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
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: $$7 < x < 17$$
Triangle Inequality Theorem Calc
Related Links:
- Worksheet on remote, exteior and inteior angles of a Triangle
- Triangle Formulas
- Triangles
- Triangle Types
- Interactive Triangle
- Remote Interior Angles
- Area of a Triangle
- Triangle Inequality Theorem
- Free Triangle Worksheets
