Triangle Inequality

Can any three lengths form a triangle? The answer is no — and the Triangle Inequality Theorem tells you exactly when three side lengths can and cannot form a triangle. This theorem appears frequently on the GED and is easy to apply once you know the rule.

What Is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle with sides a, b, and c:

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• \(\color{blue}{a + b > c}\)
• \(\color{blue}{a + c > b}\)
• \(\color{blue}{b + c > a}\)

All three conditions must hold. If any one fails, the three lengths cannot form a triangle.

How to Apply the Triangle Inequality

Quick check using the two smaller sides

In practice, the only test that can fail is the one involving the two smallest sides vs. the largest side. If the two smallest sides sum to more than the largest, all three conditions are automatically satisfied.

• Sides 3, 4, 5: smallest pair \(\color{blue}{3 + 4 = 7}\) > 5 ✓ → valid triangle
• Sides 2, 3, 6: smallest pair \(\color{blue}{2 + 3 = 5}\) < 6 → NOT a valid triangle
• Sides 5, 8, 12: \(\color{blue}{5 + 8 = 13}\) > 12 ✓ → valid triangle
• Sides 4, 4, 7: \(\color{blue}{4 + 4 = 8}\) > 7 ✓ → valid triangle
• Sides 1, 2, 10: \(\color{blue}{1 + 2 = 3}\) < 10 → NOT a valid triangle

Finding the range for an unknown third side

If two sides are known, say a and b, the third side c must satisfy:

|\(\color{blue}{a – b}\)| < c < \(\color{blue}{a + b}\)

For sides 4 and 7: \(\color{blue}{|4 – 7| = 3 < c < 11}\), so the third side must be between 3 and 11 (not including 3 or 11).

Step-by-Step Summary

1. Order the three side lengths from smallest to largest.
2. Add the two smaller sides.
3. Compare that sum to the largest side.
4. If the sum is greater than the largest side, the triangle is valid.
5. If the sum equals or is less than the largest side, no triangle can be formed.

Watch: Triangle Inequality Theorem (Video Lesson)

Khan Academy demonstrates the theorem with multiple examples and explains the geometric reasoning:

Worked Examples

Example 1: Can sides of length 3, 4, and 5 form a triangle?

\(\color{blue}{3 + 4 = 7}\) > 5 ✓; \(\color{blue}{3 + 5 = 8}\) > 4 ✓; \(\color{blue}{4 + 5 = 9}\) > 3 ✓. Yes, they form a valid triangle (and it is a right triangle).

Example 2: Can sides 2, 3, and 6 form a triangle?

\(\color{blue}{2 + 3 = 5}\) which is NOT greater than 6. No, these lengths cannot form a triangle.

Example 3: Two sides of a triangle are 4 and 7. What are the possible whole-number values for the third side?

\(\color{blue}{|4 – 7| < c < 4 + 7 \rightarrow 3 < c < 11}\). Whole-number values: 4, 5, 6, 7, 8, 9, 10.

Example 4: Sides 5, 8, and 12 — valid triangle?

\(\color{blue}{5 + 8 = 13}\) > 12 ✓. Yes, valid triangle.

More Practice: Triangle Inequality Theorem (Video)

Math with Sohn explains how the theorem works and demonstrates classifying triangle sides:

Exercises

1. Can sides 6, 8, and 14 form a triangle?
2. Can sides 7, 10, and 15 form a triangle?
3. Two sides of a triangle are 5 and 9. What is the range of possible values for the third side?
4. Two sides of a triangle are 6 and 6. What are the possible whole-number values for the third side?
5. Can sides 3, 3, and 3 form a triangle?
6. A triangle has sides 4, 4, and 9. Is this valid? Explain.

Answers

1. \(\color{blue}{6 + 8 = 14}\) which is NOT greater than 14. No (equality is not allowed).
2. \(\color{blue}{7 + 10 = 17}\) > 15. Yes.
3. |\(\color{blue}{5 – 9}\)| = 4 < c < 14. Third side: between 4 and 14 (exclusive).
4. 0 < c < 12; whole-number values: 1 through 11.
5. \(\color{blue}{3 + 3 = 6}\) > 3. Yes — equilateral triangle.
6. \(\color{blue}{4 + 4 = 8}\) < 9. No — not a valid triangle.

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Frequently Asked Questions

What happens when two sides sum exactly to the third side?

If \(\color{blue}{a + b = c}\), the three “sides” would lie in a straight line — a degenerate triangle with zero area. The theorem requires the sum to be strictly greater than the third side, so equality does not work.

Why does only the two-shortest-sides test matter?

The triangle inequality must hold for all three pairs, but the only pair that can possibly fail is the two shorter sides vs. the longest side. If the two shortest lengths sum to more than the longest, the other two inequalities are automatically satisfied.

How is the Triangle Inequality Theorem used in real life?

Engineers and architects use it when designing triangular supports — if the side lengths are chosen incorrectly, the structure cannot close. It also underpins GPS distance calculations and shortest-path problems in computer science.

Related Topics

• Triangles
• Relationship Between Sides and Angles in a Triangle
• Area of a Triangle
• Polygons
• Area of a Parallelogram