About the Triangle Inequality Checker
The triangle inequality theorem says that three lengths can form a triangle only if the sum of any two of them is greater than the third. This triangle inequality checker tests all three conditions at once and tells you plainly whether your sides make a real triangle, a degenerate “flat” triangle (when two sides add up exactly to the third), or no triangle at all.
When the sides do work, it classifies the triangle by its sides (equilateral, isosceles or scalene) and by its angles (acute, right or obtuse, using the converse of the Pythagorean theorem), and gives the three angles, the perimeter and the area by Heron’s formula. It also shows the range of lengths the third side may take given the first two.
Students use it for homework and test prep; woodworkers, welders and designers use it to check whether three pieces will close into a triangle before cutting anything.
How to use the triangle inequality checker
- 1Enter the three side lengths in the same unit.
- 2Read the verdict: triangle, degenerate or not a triangle.
- 3Check the ✓/✗ list to see which inequality fails, if any.
- 4For a valid triangle, review its type, angles, perimeter and area.
Formula and method
Three positive lengths form a triangle exactly when each is shorter than the sum of the other two. It is enough to check that the two shortest sides together exceed the longest; equivalently, the third side must lie strictly between the difference and the sum of the other two. If the shorter sides add up exactly to the longest, the “triangle” collapses into a straight line (degenerate, zero area).
For a valid triangle, comparing the squares of the sides classifies its largest angle (the converse of the Pythagorean theorem): with s₁ and s₂ the shorter sides and L the longest, s₁² + s₂² = L² means a right angle, greater means acute and smaller means obtuse. Angles come from the law of cosines and the area from Heron’s formula, √(s(s − a)(s − b)(s − c)) with s the semiperimeter.
- a, b, c
- The three side lengths
- s₁, s₂, L
- The sides sorted from shortest to longest
- s
- Semiperimeter (a + b + c) ÷ 2
Worked examples
Sides 7, 10 and 5
The two shorter sides, 5 and 7, add to 12, which beats 10, so the sides form a triangle. Since 5² + 7² = 74 is less than 10² = 100, the angle opposite the 10 side is obtuse (about 111.8°). Heron’s formula gives an area of about 16.25.
Sides 3, 4 and 8
3 + 4 = 7 is less than 8, so the two short sides cannot reach each other across the long side. Given sides of 3 and 4, the third side would need to be strictly between 1 and 7.
Sides 5, 12 and 13
This is a Pythagorean triple: 5² + 12² = 25 + 144 = 169 = 13², so the triangle is right-angled with the 90° angle opposite the 13 side. Its area is ½ × 5 × 12 = 30 and its perimeter is also 30.
Frequently asked questions
What is the triangle inequality theorem?+
It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If that fails for any pair, the three lengths cannot be assembled into a triangle.
How do I know if three numbers can be the sides of a triangle?+
Sort them and check that the two smallest add up to more than the largest. For 4, 6 and 9: 4 + 6 = 10 > 9, so they form a triangle. For 2, 3 and 6: 2 + 3 = 5 < 6, so they do not.
What is the range for the third side of a triangle?+
If two sides are a and b, the third side must be greater than |a − b| and less than a + b. For sides 7 and 10, the third side must lie strictly between 3 and 17.
What happens if two sides add up exactly to the third?+
You get a degenerate triangle: the three vertices lie on a straight line and the area is zero. Most textbooks do not count it as a triangle, which is why the inequality is strict.
How can I tell if a triangle is acute, right or obtuse from its sides?+
Compare the square of the longest side with the sum of the squares of the other two. Equal means right, smaller means acute and larger means obtuse — for example 6, 8, 11 is obtuse because 121 > 100.