Triangle Inequality

|a + b| ≤ |a| + |b| — and triangle sides obey a + b > c.

The idea

The triangle inequality appears in three forms.

Theorem (Triangle inequality).

For vectors, $|\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|$. For real numbers, $|a + b| \leq |a| + |b|$. For the side lengths $a, b, c$ of a triangle, each side is strictly less than the sum of the other two: $a + b > c$, $a + c > b$, and $b + c > a$.

The three forms state one fact. Lay $\vec{a}$ and $\vec{b}$ tip-to-tail; the two arrows and their sum $\vec{a} + \vec{b}$ close up into a triangle, in which $|\vec{a} + \vec{b}|$ is the direct side and $|\vec{a}| + |\vec{b}|$ is the detour through the corner, and the direct path is never longer than the detour. The real numbers are the case where both vectors lie on one line, and there equality holds exactly when the numbers share a sign.

In a genuine triangle the sides do not lie on one line, so the inequality is strict, and it bounds an unknown third side from both ends: given sides $a$ and $b$, a third side $x$ must satisfy $|a - b| < x < a + b$.

One further ordering fact: in any triangle, the largest angle lies opposite the longest side, and the smallest angle opposite the shortest.

Ways to work on it

• Walkthrough. Numbers, the triangle test, the third-side range, and side–angle order.
• Practice. Find the minimum third side — uses |a - b| bound.
• Hardest. Max third side, count of integer sides, or the largest angle.

Not sure where to start? Take the ten-question placement test.