Triangle Inequality Theorem
Triangle Inequality Theorem helps us decide whether three side lengths can actually make a triangle.
This idea is very important because not every set of three lengths can connect to form a closed shape. Sometimes the sides are too short or too long compared to each other, so they cannot meet.
In this lesson, you will learn what the Triangle Inequality Theorem says, how to test side lengths, and how to solve problems using it.
What is the Triangle Inequality Theorem?
The Triangle Inequality Theorem says that in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
If a triangle has side lengths \(a\), \(b\), and \(c\), then all three of these must be true:
$$a+b>c$$
$$a+c>b$$
$$b+c>a$$
If even one of these is not true, then the three lengths cannot form a triangle.
Why does this make sense?
Imagine trying to connect two short sides to reach across a very long third side. If the two short sides together are not longer than the third side, they cannot bend enough to close the shape.
For example, if you had lengths 2, 3, and 10, the two smaller sides add to \(2+3=5\). Since 5 is less than 10, they cannot reach far enough to connect. So no triangle can be made.
An easy way to check
You can test all three inequalities, but there is also a shortcut.
If you put the side lengths in order from smallest to largest, you really only need to check whether the sum of the two smaller sides is greater than the largest side.
So if the sides are \(x\le y\le z\), check:
$$x+y>z$$
If this is true, the other two inequalities will also be true.
Important detail
The theorem says greater than, not greater than or equal to.
If the sum of two sides equals the third side, the sides form a straight line, not a triangle.
For example:
$$4+5=9$$
Since the sum is exactly equal to the third side, these lengths do not form a triangle.
Steps for checking side lengths
Identify the three side lengths.
Find the largest side.
Add the two smaller sides.
Compare that sum to the largest side.
If the sum is greater, a triangle can be formed. If not, it cannot.
Worked Example 1: Can these lengths form a triangle?
Side lengths: 5, 7, 9
The largest side is 9. Add the two smaller sides:
$$5+7=12$$
Now compare:
$$12>9$$
Since the sum of the two smaller sides is greater than the largest side, these side lengths can form a triangle.
Worked Example 2: Can these lengths form a triangle?
Side lengths: 3, 4, 8
The largest side is 8. Add the two smaller sides:
$$3+4=7$$
Compare:
$$7<8$$
Since the sum of the two smaller sides is less than the largest side, these side lengths cannot form a triangle.
Worked Example 3: What if the sum is equal?
Side lengths: 6, 2, 4
The largest side is 6. Add the two smaller sides:
$$2+4=6$$
Compare:
$$6=6$$
This is not greater than 6. So these lengths do not form a triangle.
They would lie flat in a straight line instead of making a closed shape.
Worked Example 4: Find the possible value of a missing side
A triangle has side lengths 8, 11, and \(x\). What values can \(x\) have?
To form a triangle, the third side must be greater than the difference of the other two sides and less than their sum.
First find the difference:
$$11-8=3$$
Then find the sum:
$$11+8=19$$
So \(x\) must satisfy:
$$3<x<19$$
This means \(x\) must be greater than 3 and less than 19.
For example, 4, 10, and 18 would all work, but 3 and 19 would not.
Why does the missing side rule work?
If one side is unknown, it still has to fit with the Triangle Inequality Theorem.
For sides 8, 11, and \(x\), the inequalities are:
$$8+11>x$$
$$8+x>11$$
$$11+x>8$$
Simplifying gives:
$$19>x$$
$$x>3$$
$$x>-3$$
The condition \(x>-3\) is always true for a side length, since side lengths are positive. So the important result is:
$$3<x<19$$
Common mistakes to avoid
Using equals instead of greater than: If the sum equals the third side, it is not a triangle.
Checking only random pairs: Be sure you compare the two smaller sides to the largest side.
Forgetting which side is largest: Always identify the biggest number first.
Including impossible side lengths: A side length must be positive.
Quick practice ideas
Ask yourself these questions when you see three side lengths:
Which side is the largest?
Do the two smaller sides add to more than that largest side?
If there is a missing side, is it between the difference and the sum of the other two sides?
Summary
The Triangle Inequality Theorem tells us whether three side lengths can form a triangle.
For a triangle to exist, the sum of any two sides must be greater than the third side. In practice, this means the two smaller sides must add to more than the largest side.
If the sum is less than or equal to the largest side, no triangle can be formed. For a missing side, its length must be greater than the difference and less than the sum of the other two sides.
Put what you read to the test
You've worked through Triangle Inequality Theorem. Try answering a few questions to see what stuck — and what might deserve a quick reread before you move on.