Math
Triangle Inequality Theorem Calculator
The triangle inequality theorem says any two sides of a triangle must add up to more than the third side.
Enter three lengths to test all three inequalities at once, or enter two sides to get the open range the third side must fall in. When the sides pass, you also get the angles and area.
Try an example
Result
Verdict
—
- a + b > c
- —
- a + c > b
- —
- b + c > a
- —
- If it is a triangle
- —
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What the triangle inequality theorem says
To close a triangle, the two shorter sides must be able to reach each other over the longest side. That happens only when their sum is strictly greater than the longest side. Written for all three sides, the theorem is three inequalities, but only the one with the longest side on the right can ever fail.
| Sides | Shorter two | Longest | Triangle? |
|---|---|---|---|
| 3, 4, 5 | 3 + 4 = 7 | 5 | Yes |
| 4, 7, 10 | 4 + 7 = 11 | 10 | Yes |
| 2, 3, 5 | 2 + 3 = 5 | 5 | No — flat (degenerate) |
| 1, 2, 10 | 1 + 2 = 3 | 10 | No |
Finding the third side
With two sides known, the third side must be shorter than their sum and longer than their difference. For sides 5 and 8 the third side is between 3 and 13, not including 3 or 13 themselves. If the sides must be whole numbers, that leaves 4 through 12.
How to use it
- Choose Check three sides or Find the range for the third side.
- Enter the lengths in any unit (all the same unit).
- Read the verdict and see which inequality passes or fails.
- If the sides pass, use the angles and area shown, or open the full triangle calculator for more.
How to read the answer
Passes means all three inequalities hold and a real triangle exists. Fails names the inequality that breaks. A range answer uses strict inequalities: the endpoints themselves give a flat triangle and are excluded.
Common mistakes and edge cases
- Testing only one inequality. Test the one with the longest side, or all three to be safe.
- Accepting equality. 2 + 3 = 5 does not pass; the sides must add to strictly more.
- Forgetting the lower bound for the third side. c must be greater than |a − b|, not just less than a + b.
- Negative or zero lengths are not sides at all.
Worked examples
Sides 4, 7, 10
Passes
Verdict
Yes — these sides form a triangle
Sides 3, 4, 5
Right triangle
Verdict
Yes — these sides form a triangle
Sides 2, 3, 5
Degenerate
Verdict
No — degenerate (flat) triangle
Sides 1, 2, 10
Fails
Verdict
No — not a triangle
Third side range for 5 and 8
3 < c < 13
Verdict
3 < c < 13
Third side range for 6 and 6
0 < c < 12
Verdict
0 < c < 12
Decimal sides 2.5, 2.5, 4.9
Verdict
Yes — these sides form a triangle
Zero side
Verdict
—
Frequently asked questions
What is the triangle inequality theorem?+
The sum of any two sides of a triangle is greater than the third side. If that fails for any pair, the lengths cannot form a triangle.
Do I need to check all three inequalities?+
Only the one where the longest side stands alone can fail, so checking shorter + middle > longest is enough. The calculator shows all three for completeness.
What if two sides add to exactly the third?+
That is a degenerate triangle: the three points are collinear, the area is zero, and it does not count as a triangle.
How do I find the possible lengths of the third side?+
It must be greater than the difference of the two known sides and less than their sum: |a − b| < c < a + b.
Does the theorem apply to angles too?+
Not directly, but the largest angle is always opposite the longest side, and the angles must sum to 180°.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026