Math Resources Hub

Triangle Inequality

Not every trio of lengths makes a triangle. Grab three sticks and the question is whether the two shorter ones can stretch far enough to meet over the longest one. If their combined length falls short, there is a gap they can never close.

7656 + 5 = 11 > 7 ✓
Sides 7, 6, and 5. The two shorter sides reach across the longest with room to spare — 6 + 5 = 11, comfortably more than 7 — so the triangle closes.
4 + 3 = 7 < 9 — no triangle439
A base of 9 with sticks of 4 and 3. Since 4 + 3 = 7 is less than 9, the sticks fall short of each other — no triangle can form.

Explore

Set the three side lengths and watch whether the triangle closes or collapses to a flat line. Push two of the sticks short while the third stays long. When exactly does the triangle fail to form?

Set the three stick lengths and see whether they close. Then press “Fold the sides up” to watch the two shorter sticks try to meet.

4
6
9

10 > 9, so sticks of length 4, 6, and 9 close into a triangle.

Three sticks of lengths a = 4, b = 6, and c = 9. They form a triangle exactly when the two shorter sticks together exceed the longest. Here the two shorter sticks sum to 10 and the longest is 9, so the figure closes into a triangle. Press “Fold the sides up” to watch the two legs swing up and meet.

Make a prediction

Make a prediction: Can three sticks of length 3, 4, and 8 form a triangle?

Test it

In the tool, take a valid triangle and shrink one side toward the difference of the other two. Right when the two shorter sides just add up to the longest, the triangle flattens into a straight segment — a "degenerate" triangle with no area. Below that, it can't close at all. That tipping point is the at work.

Why it works

The shortest path between two points is a straight line. To travel from one end of the longest side to the other, you can go directly along that side, or you can detour through the third vertex — along the other two sides. The detour can never be shorter than the straight route, so the other two sides together must be longer than the longest side. Written out for sides aa, bb, cc:

a+b>c,a+c>b,b+c>aa + b > c, \qquad a + c > b, \qquad b + c > a

In practice you only need to check the largest side against the sum of the other two — if that one passes, the rest automatically do.

Key result

Three lengths form a triangle exactly when every pair sums to more than the third side — equivalently, when the longest side is shorter than the other two combined. With two sides aa and bb fixed, the third must land strictly between ∣a−b∣|a - b| and a+ba + b.

Worked example: Two sides of a triangle are 7 and 10. Find the full range of possible lengths for the third side.

  1. The third side has to be shorter than the other two combined: 7+10=177 + 10 = 17.

    Before the next step: The third side must be less than…

Practice

Score: 0/4

  1. 1.Which set of three lengths can form a triangle?
  2. 2.Two sides of a triangle are 5 and 8. The third side must be less than what whole number?
  3. 3.Two sides of a triangle are 5 and 8. The third side must be greater than what whole number?
  4. 4.Can sides of length 4, 6, and 10 form a triangle?
For teachers

Standard, common misconceptions, and suggested use will live here.