Require a strict inequality
Three lengths form a nondegenerate triangle when each is positive and every pair sums to more than the remaining length. Equality makes the figure flat. If one pair sums to less than the third, the shorter segments cannot meet across the longest segment. This is an existence test for side lengths, not a classification of the angles. Apply it before using inverse cosine or an area formula, because later numerical outputs cannot make inconsistent side data describe a valid triangle.
Sort the lengths for a shorter check
Let x≤y≤z be the positive lengths in ascending order. Then it is sufficient to test x+y>z, because the other pair sums necessarily exceed the smaller remaining side. First verify positivity: a zero or negative input invalidates the reasoning that makes this shortcut work. Sorting changes the order used for the test but does not change any original side-angle labels in a later triangle solution. Preserve the original labels if you will associate sides with opposite angles.
Compare valid, flat and impossible examples
Sides 4,7,9 pass because 4+7=11>9. Sides 4,5,9 are exactly flat because 4+5=9. Sides 4,5,10 fail because 9<10. The difference between strict greater-than and greater-than-or-equal-to matters: including equality would count a zero-area segment arrangement as an ordinary triangle. A sketch can illustrate closure, but lengths rather than the visual shape of an approximate drawing decide this particular mathematical test.
Derive the allowed third-side range
If two sides are a and b and the third is c, combine the three strict inequalities to obtain |a−b|<c<a+b. With sides 7 and 10, the allowed range is 3<c<17. Both endpoints are excluded. A third side of 3 or 17 is flat, 2 is too short and 18 too long. Every positive value strictly inside the interval gives a triangle mathematically, though very extreme ratios can be numerically difficult in a particular tool.
| Candidate c | Test | Conclusion |
|---|---|---|
| 2 | Below 3 | Impossible |
| 3 | Lower endpoint | Flat |
| 8 | 3 < 8 < 17 | Valid |
| 17 | Upper endpoint | Flat |
| 18 | Above 17 | Impossible |
Endpoint inequalities are strict for positive-area triangles.
Distinguish existence from a unique solution
Knowing only sides 7 and 10 does not determine c; it determines an interval of possible values. Different allowed c values give different shapes and areas. Three valid SSS measurements, by contrast, determine one triangle up to congruence and reflection. Three angles can establish shape but not absolute size. These are separate sufficiency questions. An existence check should not be presented as a complete solution when some required side or angle remains unspecified.
Do not confuse the test with Pythagoras
After existence is established, comparing z² with x²+y² can classify an angle under the appropriate longest-side interpretation. That is a different calculation. The side sum x+y>z establishes closure; a sum of squares does not replace it. A valid triangle need not be right-angled. Sides 4,7,9 satisfy the triangle inequality but do not satisfy a right-triangle equality. Use the general Triangle Calculator rather than assuming Pythagoras is a universal relationship among three sides.
Consider measurement uncertainty near an endpoint
Suppose recorded sides are 5.0,5.0,10.0 after rounding to tenths. The recorded numbers are degenerate, but their underlying unrounded values might lie on either side of closure. The rounded record alone does not establish a small positive area. Obtain more precise measurements or report the uncertainty rather than adding an arbitrary epsilon to force acceptance. The calculator's numerical safeguards also reject some near-flat cases where a reliable angle and area cannot be resolved within its supported precision.
Use the calculator as a transparent check
Select SSS and enter 4,7,9 for a valid example. Enter 4,5,9 to verify that a flat case is not reported as an ordinary positive triangle, and 4,5,10 for a no-solution case. Keep the tool's explanation with the result. Invalid measurements and numerical limitations should not be interpreted as a physical diagnosis. The interface uses positive side inputs and does not correct units, infer intended dimensions or decide which measurement should be replaced.
Document a useful validation result
A concise record lists the three source lengths in a common unit, the sorted values, the longest-side inequality and whether it is strict. If one side is being selected, state the open interval and any separate design restrictions. Converting all sides by the same positive factor preserves validity, so centimeters and meters should give consistent closure conclusions after proper conversion. More displayed digits do not justify a claim of higher measurement accuracy. Keep geometric existence, numerical resolvability and practical fit as distinct conclusions.