Triangle Inequality Checker
Check whether three side lengths can form a valid triangle. Enter the values for sides a, b, and c, and the calculator applies the Triangle Inequality Theorem to tell you if a triangle is possible.
Check the Triangle Inequality
Triangle Inequality Theorem Formula
Where:
- \(a\), \(b\), \(c\) — the lengths of the three sides
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side. All three inequalities must hold; if even one fails, the sides cannot form a triangle.
Triangle Inequality – Calculation Example
Example: check whether the sides \(a = 5\), \(b = 6\), and \(c = 10\) can form a triangle.
\[a + b = 5 + 6 = 11 > 10 \;\checkmark\]
\[a + c = 5 + 10 = 15 > 6 \;\checkmark\]
\[b + c = 6 + 10 = 16 > 5 \;\checkmark\]
Result: all three inequalities are satisfied, so a triangle with sides 5, 6, and 10 is possible.
How to Use the Calculator
- Enter the length of Side a in the first field.
- Enter the length of Side b in the second field.
- Enter the length of Side c in the third field.
- Read the Triangle Inequality Result: the calculator tells you whether a triangle with those three sides is possible.
All side lengths must be positive numbers. If any of the three inequalities fails, the result will state that a triangle is not possible.
What Is the Triangle Inequality Theorem?
The Triangle Inequality Theorem is one of the fundamental rules of Euclidean geometry. It states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Intuitively, this reflects the fact that a straight line is the shortest path between two points: if one side were as long as or longer than the sum of the other two, the two shorter sides would have to lie flat along the third side, and the figure would collapse into a straight line segment rather than a triangle.
It is enough to verify the single largest side against the sum of the two smaller ones. If the largest side is shorter than the sum of the other two, the remaining two inequalities hold automatically. The calculator checks all three conditions, which is a robust way to test sides regardless of which one is largest.
The theorem has practical uses beyond geometry class. Engineers and architects use it when verifying that structural elements can close into a triangular frame; surveyors rely on it when checking that measured distances are consistent; and programmers use it in computer graphics and physics engines to validate geometric data. It also underpins the definition of distance in mathematics: any valid distance function must satisfy a triangle inequality of its own.
Note that the theorem only checks feasibility. It tells you that a triangle exists, but not its angles or area. If you need those, the sides determine the triangle completely, so you can go on to compute its angles with the Law of Cosines or construct auxiliary lines such as an angle bisector or the length of an angle bisector.
Assumptions, Limitations and Tips
- Strict inequality: if the sum of two sides equals the third (for example, 3, 4, and 7), the result is a degenerate "triangle" — a straight line. The theorem requires a strict greater than, so such input is reported as impossible.
- Positive lengths only: side lengths must be positive real numbers. Zero or negative values have no geometric meaning for a triangle.
- Equal sides are valid: an equilateral triangle (e.g., 5, 5, 5) or an isosceles triangle (e.g., 5, 5, 8) passes the test as long as the inequalities hold.
- Units: the calculator works with any unit of length (cm, inches, meters), but all three sides must use the same unit for the comparison to be meaningful.
- It does not compute angles or area: if you also need the triangle's dimensions or related vectors, see tools like the 2D Vector Magnitude calculator or the 2D Vector Angle calculator for follow-up computations.
Frequently Asked Questions
How can I check the triangle inequality by hand without a calculator?
Find the longest of the three sides and compare it with the sum of the other two. If the longest side is strictly smaller than that sum, a triangle is possible; otherwise it is not. For example, with sides 4, 9, and 14: the longest side is 14 and \(4 + 9 = 13 < 14\), so no triangle exists. Only one comparison is needed because the other two inequalities hold automatically.
What happens if the sum of two sides exactly equals the third?
You get a degenerate case: the three segments lie flat on a straight line instead of forming a closed shape with area. For sides like 3, 4, and 7, we have \(3 + 4 = 7\), so the triangle inequality fails. The theorem requires a strict inequality \(a + b > c\), and this checker correctly reports such input as impossible.
Can the sides be in different units, like 5 inches, 30 cm, and 1 meter?
No. All three lengths must be converted to the same unit before testing, otherwise the comparison is meaningless. Convert first: 5 inches ≈ 12.7 cm and 1 meter = 100 cm, giving sides 12.7, 30, and 100. Then check: \(12.7 + 30 = 42.7 < 100\), so these lengths cannot form a triangle regardless of unit choice.
Do the sides need to be in any particular order when I enter them?
No, the order does not matter. The calculator tests all three inequalities \(a + b > c\), \(a + c > b\), and \(b + c > a\), so it works no matter which value you type into which field. The triangle is the same geometric figure whether its sides are listed as 5, 6, 10 or 10, 5, 6.
Can I use decimals or fractions for the side lengths?
Yes. Side lengths can be any positive real numbers, including decimals such as 2.5, 3.75, or 4.1. The inequality works the same way: for 2.5, 3.75, and 6.2, check \(2.5 + 3.75 = 6.25 > 6.2\), so a triangle is possible — though a very flat one, since the sides barely satisfy the condition.
What is the difference between the triangle inequality and the Pythagorean theorem?
The triangle inequality only asks whether three lengths can close into any triangle at all; it says nothing about angles. The Pythagorean theorem is stricter and applies only to right triangles, requiring \(a^2 + b^2 = c^2\). A set of sides can easily pass the inequality test without being a right triangle — 5, 6, and 10 form a valid triangle but not a right one.
If the test passes, how do I find the angles of my triangle?
Use the Law of Cosines. Once you know a triangle is possible, its angles are fully determined by the sides. For example, the angle \(C\) opposite side \(c\) is \(\cos C = \frac{a^2 + b^2 - c^2}{2ab}\). A rough check: if \(c^2 < a^2 + b^2\) the triangle is acute, if equal it is right, and if greater it is obtuse.