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Instant Calculation
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The triangle inequality theorem states that for three side lengths to form a valid, real triangle, the sum of any two of those sides must be strictly greater than the third. It's a deceptively simple check with real practical weight: before doing any further geometry (finding angles, area, or classifying the shape), you first need to confirm the three numbers you were given can actually close up into a triangle at all — three arbitrary positive numbers do not automatically form one. This calculator checks all three necessary combinations at once.
How it works
Enter three side lengths, a, b, and c. The calculator checks all three pairwise combinations: a+b>c, b+c>a, and a+c>b, reporting pass or fail for each individually plus an overall verdict. Intuitively, each side represents a straight path between two triangle vertices, and if the two shorter sides combined can't reach as far as the longest side alone, the three segments simply can't meet up to enclose any area — they'd either fall short of connecting or, in the boundary case, lie perfectly flat on top of each other as a degenerate straight line.
- Enter side a.
- Enter side b.
- Enter side c.
- Click Calculate to see your results.
Examples
3, 4, 5
All three checks pass: 3+4=7>5, 4+5=9>3, and 3+5=8>4, so this is a valid triangle (in fact, a well-known right triangle).
1, 1, 10
1+1=2, which is not greater than 10, so this check fails immediately — the two short sides can't possibly reach far enough to meet the long side, so no triangle exists.
5, 5, 10
5+5=10, which equals but does not exceed the third side — this fails the STRICT inequality, producing a degenerate "triangle" that's actually just a flat straight line with zero area, not a real triangle.
Who should use it
- Verifying three measured or given lengths can actually form a triangle before further calculations.
- Geometry coursework on the triangle inequality theorem.
- Sanity-checking survey or construction measurements.
Industry applications
- Geometry education
- Surveying and construction (measurement validation)
Advantages
- Fast, definitive validity check before any further triangle math.
- Checks all three combinations for full transparency.
Limitations
- Only confirms a valid triangle exists — doesn't classify its shape or compute its properties.
Common mistakes to avoid
- Only checking one combination of sides instead of confirming the smallest-two-against-largest comparison covers the strictest case.
- Treating an equality case (sum exactly equals the third side) as valid — it fails the strict inequality and produces a degenerate, zero-area "triangle."
- Mixing measurement units between the three side lengths before checking.
Best practices
- Sort your three lengths from smallest to largest first, then just check whether the two smallest sum to more than the largest — that single check is sufficient.
- Treat an equality result (sum exactly equals the third side) as a failure, not a valid degenerate triangle.
- Run this check before attempting any further triangle calculation (area, angles, classification) on a given set of three lengths.
Tips
- Once you've confirmed a valid triangle, use the Classifying Triangles Calculator to determine its side and angle type.