Triangle Inequality Theorem Calculator
Check whether three side lengths can form a triangle with the triangle inequality theorem, or find the possible range of the third side from two sides, plus the triangle type and area.
How it works
Three lengths a, b, c form a triangle ⇔ a + b > c, a + c > b and b + c > a (enough to check: the two shorter sides add to more than the longest) · third side x from a and b: |a − b| < x < a + b
The triangle inequality theorem says the sum of any two sides of a triangle is greater than the third side, because the straight path between two vertices is shorter than the detour through the third vertex. If the sum equals the third side the “triangle” collapses into a segment (degenerate). Rearranging the inequalities gives the range for an unknown third side: strictly between the difference and the sum of the other two (Euclid, Elements I.20; Khan Academy, Triangle inequality theorem). For a valid triangle the calculator also classifies it and finds its area with Heron’s formula.
Worked example
Sides 5, 7 and 10: 5 + 7 = 12 > 10, 5 + 10 = 15 > 7 and 7 + 10 = 17 > 5, so a triangle can be formed. It is scalene, and since 5² + 7² = 74 < 10² = 100 it is obtuse. Perimeter 22; Heron: s = 11, area = √(11 · 6 · 4 · 1) = √264 ≈ 16.248077. With only sides 5 and 7, the third side must satisfy 2 < x < 12.
Sources: Khan Academy — Triangle inequality theorem · Wolfram MathWorld — Triangle Inequality
Frequently asked questions
What is the triangle inequality theorem?
In any triangle, the sum of the lengths of any two sides is greater than the length of the remaining side.
Can 3, 4 and 8 be the sides of a triangle?
No. 3 + 4 = 7, which is less than 8, so the two short sides cannot reach each other.
How do you find the range of the third side?
Subtract and add the two known sides: the third side must be greater than their difference and less than their sum. For 5 and 7: 2 < x < 12.
Do I really need to check all three inequalities?
Only one matters in practice: if the two shortest sides add up to more than the longest side, the other two inequalities hold automatically.
What happens when the sum equals the third side?
You get a degenerate triangle: the three points lie on one line and the area is 0. Most textbooks do not count it as a triangle.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.