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Glossary

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Triangles and TrigonometryThe Triangle Inequality

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Having studied many of the properties and components of triangles, let’s think about creating triangles. In particular, if I give you any three numbers, can you make a triangle that has those side lengths?

Here are a some examples – move the vertices of the triangle until the three sides match one of the triples on the left.

5
6
7
3
9
9
2
4
8
4
6
7
1
2
6
3
5
7

It seems like there are a few cases where three numbers simply cannot make a triangle. This particularly happens when one side is much longer thanis much shorter thanis the same length as the other two.

Think about the three sides of a triangle as metal rods, connected with hinges. Let’s place the longest rod in the middle and the shorter ones on either side.

Now it is easy to see that it is impossible to link up the ends of the shorter rods, if their combined length is less than the length of the larger rod.

Let’s rewrite this observation in mathematical terms:

The Triangle Inequality
The sum of the lengths of any two sides of a triangle must be greater than the length of the third.

In other words, if a triangle has sides a, b and c, then we know that a+b>c and a+c>b and b+c>a.

The triangle inequality allows us to quickly check if three numbers can make a triangle. Which of these triples of numbers are possible?

4
6
9
1
2
3
3
7
8
2
4
7
1
5
8
2
3
4

The triangle inequality also allows us to estimate the length of the third side of a triangle, if we know the length of the other two.

Imagine that a triangle has two sides of length 4 and 6. Let’s call c the length of the third side. Then we know that

4+6>c, 4+c>6 and 6+c>4

We can rearrange these inequalities to give <c< . The length of side c has to be between 2 and 10.

Once again, we can think about this using physical objects: two sides of the triangle are metal rods of length 4 and 6, and the third side is a rubber band that can expand or contract.

Now you can see that the rubber band will always be longer than 64=2 and shorter than 6+4=10.

Note that these are strict inequalities. If the third side is exactly 2 or 10, we get a straight line and not a triangle. However 2.1 or 9.9 would be enough to form a triangle.