Triangle Inequality Calculator
Check whether three positive side lengths can form a non-degenerate triangle, identify the tightest inequality, and export the complete calculation to a validated Excel workbook.
Side lengths
Enter positive lengths. Mixed units are converted automatically.
Live result
All comparisons use the unit selected for side a.
All three strict inequalities are satisfied.
Inequality checks
Values shown in ft
| Rule | Left side | Relation | Right side | Margin | Status |
|---|---|---|---|---|---|
| a + b > c | 10.5 ft | > | 9 ft | +1.5 ft | Pass |
| a + c > b | 13.5 ft | > | 6 ft | +7.5 ft | Pass |
| b + c > a | 15 ft | > | 4.5 ft | +10.5 ft | Pass |
How to use the triangle inequality calculator
What this calculator does
This calculator determines whether three positive line segments can be the sides of a non-degenerate Euclidean triangle. It checks the three strict conditions a + b > c, a + c > b, and b + c > a. The result is an exact logical test of the entered lengths, subject only to normal floating-point precision during unit conversion. It does not determine the triangle's angles, orientation, area, structural strength, or whether a physical construction is safe.
When to use it
Use the tool when checking geometry homework, screening three measured parts before fabrication, validating survey or CAD inputs, or confirming that three proposed edge lengths can close into a triangular frame. It is especially useful near the boundary where the two shorter sides almost equal the longest side. For the underlying theorem and its equivalent forms, see the University of Texas at San Antonio's triangle inequality mathematics reference.
How to calculate
- The calculator opens with a complete demonstration: side a = 4.5 ft, side b = 6 ft, and side c = 9 ft. The live result and a validated example Excel workbook are immediately available.
- Replace each Length value under Side a, Side b, and Side c. Use positive decimal numbers with a period as the decimal separator. Standard U.S. thousands grouping such as 1,250.5 is accepted; scientific notation and decimal commas are rejected.
- Choose the matching Unit for each side. You may mix millimeters, centimeters, meters, inches, feet, and abstract units. Changing a unit converts the current number so the physical length remains the same. All displayed comparisons use side a's selected unit.
- Read Triangle status first, then inspect Checks passed, Critical margin, Longest side, and Perimeter. Review the Inequality checks table to see exactly which rule passes or fails.
- Select Download Excel to export the current inputs, converted values, outputs, and all three checks as a real .xlsx workbook. Select Reset to clear the demonstration and results. Reset may disable Excel export until all three positive lengths are entered again.
Input guide
Side a – Length, Side b – Length, and Side c – Length are required positive decimal values. Each may be no more than 1 trillion in its selected unit. Examples are 4.5, 6, and 9. Larger values increase sums and margins, but validity depends on proportions rather than overall scale. Zero, negative numbers, words, unsupported symbols, malformed grouping, and values that overflow after conversion are rejected. A common mistake is assuming equality is enough: 3, 6, and 9 lie on a straight line and do not form a proper triangle.
Each side's Unit is required and may be mm, cm, m, in, ft, or units. The first five choices are physical lengths and are converted through meters. The units option is dimensionless and should be used only when all three entries share the same abstract scale; it cannot be mixed meaningfully with physical units, so the calculator rejects that combination. Switching a unit converts the current valid entry rather than merely relabeling it. Avoid typing unit symbols inside the Length field.
Output guide
Triangle status is the primary exact conclusion: Valid triangle, Degenerate triangle, or Not a triangle. Checks passed counts successful strict inequalities out of three. Critical margin is the smallest left-minus-right difference; positive means valid, zero marks the degenerate boundary, and negative measures the shortfall. Longest side identifies the largest entered segment after conversion. Perimeter is shown only for a valid triangle; otherwise it is not applicable because the lengths do not bound a proper triangular region.
The header pills repeat the status, pass count, and output unit for quick scanning. In the Inequality checks table, Rule names the tested inequality, Left side is the sum of two sides, Relation shows >, =, or <, Right side is the remaining side, Margin is Left side minus Right side, and Status is Pass or Fail. Every value is driven by the three lengths and their units; the table is an exact comparison, not a probability or recommendation.
Worked example
With a = 4.5 ft, b = 6 ft, and c = 9 ft, the checks are 4.5 + 6 = 10.5 > 9, 4.5 + 9 = 13.5 > 6, and 6 + 9 = 15 > 4.5. All three pass, so the first-open result is Valid triangle. The smallest margin is 10.5 – 9 = 1.5 ft, side c is longest at 9 ft, and the perimeter is 4.5 + 6 + 9 = 19.5 ft.
The classical statement appears as Proposition 20 in Euclid's Elements, Book I: the sum of any two sides is greater than the remaining side.
How the formula works
For positive lengths, it is enough to sort the values from shortest to longest and test whether the two shorter sides add to more than the longest. The calculator still displays all three inequalities because that makes the failure mode explicit and preserves the standard theorem statement.
Valid triangle ⇔ min(a + b – c, a + c – b, b + c – a) > 0The minimum of those three differences is the critical margin. Scaling every side by the same positive factor scales the margin and perimeter by that factor but does not change validity. This is why 3-4-5 and 30-40-50 have the same triangle shape and both pass.
Boundary cases and interpretation
If the critical margin is exactly zero, the points are collinear: the shape has zero area and is classified as degenerate. A negative margin means the two relevant sides are too short to meet across the remaining side. The magnitude of the negative margin is the minimum extra length needed in that comparison, assuming the other two lengths stay fixed.
The broader vector and norm interpretation is covered by Wolfram MathWorld's triangle inequality overview. For ordinary side-length screening, use the strict geometric rule shown here and keep all three inputs in compatible units.