Triangle inequality theorem | Geometry | Quantitative reasoning

The Triangle Inequality Theorem helps you find the minimum and maximum possible lengths of a missing side in any triangle. You may also see it called the Third Side Rule or the 3rd Side Rule.

Don’t confuse this with the Pythagorean Theorem, which you can use to find the exact length of a missing side in a right triangle.

Consider the triangle below, with two sides of $4$ and $9$ .

Example of triangle inequality theorem with two sides given and one missing side

We haven’t stated the rule yet, but pause and think: what lengths could the missing side have?

(spoiler)

The missing side must be in the range of $5 < x < 13$

Even though we don’t know any angles, we can still pin down a range for the third side. That’s exactly what the Triangle Inequality Theorem gives you.

Definitions

Triangle inequality theorem: $(a - b) < c < (a + b)$

Triangle with labeled sides a b c

In words, the third side must satisfy both conditions:

Greater than the difference of the other two sides
Less than the sum of the other two sides

Let’s return to our example triangle.

Example of triangle inequality theorem with two sides given and one missing side

Applying the theorem, the missing side $x$ must be:

Greater than $9 - 4 = 5$

This lower bound is easiest to picture. The smallest possible third side happens when the angle between the $4$ and $9$ sides is very small - almost $0^{\circ}$ - so the two sides nearly lie on top of each other. If the missing side were smaller than $5$ , the endpoints wouldn’t meet (there would be a gap). If the missing side were exactly $5$ , the “triangle” would collapse into a straight line segment, not a triangle.
Less than $9 + 4 = 13$

The largest possible third side happens when the angle between the $4$ and $9$ sides is as large as possible - almost $18 0^{\circ}$ - so the two sides are nearly in a straight line end-to-end. If the missing side were longer than $13$ , it would be too long to connect the endpoints. If it were exactly $13$ , you’d again get a straight line, not a triangle.

So $x$ must be greater than $5$ and less than $13$ , meaning $5 < x < 13$ .

Sidenote

A straight line isn't a triangle

This is obvious, but it’s a useful way to remember why the inequalities are strict. Look at the figure below.

Impossible triangle because of the third side rule

Does it look like a normal triangle? Look again.

This triangle is impossible: the sides aren’t long enough to connect. You’d be left with a gap of $2$ .

Impossible triangle because of the third side rule on a line

Remember, figures on the GRE are not drawn to scale.

A GRE question may ask directly for the range of a missing side, but it may also ask something one step removed, like the perimeter.

Which of the following could be the perimeter of the triangle shown in the figure?

Example of triangle inequality theorem with two sides given and one missing side

Select all that apply.
A. 9
B. 10
C. 12
D. 14
E. 17

Can you apply the Triangle Inequality Theorem to answer this?

(spoiler)

Answer: C and D

First, find the possible range of the missing side $x$ . The difference of the known sides is $5 - 3 = 2$ , and their sum is $5 + 3 = 8$ . So the third side must satisfy:

$2 < x < 8$

Now translate that into a range for the perimeter $p$ . The two known sides add to $5 + 3 = 8$ , so:

Since $x > 2$ , the perimeter must be greater than $5 + 3 + 2 = 10$ .
Since $x < 8$ , the perimeter must be less than $5 + 3 + 8 = 16$ .

So the perimeter must satisfy:

$10 < p < 16$

Notice that this range is exclusive: it does not include $10$ or $16$ . If $x$ were exactly $2$ or exactly $8$ , the figure would collapse into a straight line rather than form a triangle.