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Triangle Inequality Calculator

Check triangle validity or compute the allowable range for a missing side using the triangle inequality theorem — fast and accurate.

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Inputs

Calculation Mode

Side a

Side b

Side c (optional for range modes)

Triangle Inequality Result

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Triangle Inequality Result—

The formula

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What Is the Triangle Inequality Theorem?

The Triangle Inequality Theorem is one of the most fundamental results in Euclidean geometry. It states that for any triangle with side lengths a, b, and c, the sum of any two sides must be strictly greater than the third side. This condition guarantees that three given lengths can actually close into a valid triangle rather than collapsing into a straight line or failing to meet at all.

The Core Formula

Three simultaneous inequalities must all hold true for a set of side lengths to form a valid triangle:

a + b > c
a + c > b
b + c > a

An equivalent compact form combines all three conditions into a single range expression: |a − b| < c < a + b. This formulation is especially useful when computing the allowable range of an unknown third side given two known sides.

Deriving the Bounds for a Third Side

When two side lengths are known and the third is unknown, the theorem directly yields the allowable open interval. Given sides a = 5 and b = 8, the third side c must satisfy |5 − 8| < c < 5 + 8, which simplifies to 3 < c < 13. Any value of c strictly between 3 and 13 produces a valid triangle with those two known sides. Values at the endpoints — c = 3 or c = 13 — produce degenerate collinear configurations, not true triangles, because the three points lie on a single straight line.

Checking Triangle Validity

To verify that three given lengths form a valid triangle, all three inequalities must be satisfied simultaneously. For example, sides 7, 10, and 15 pass the test: 7 + 10 = 17 > 15, 7 + 15 = 22 > 10, and 10 + 15 = 25 > 7. By contrast, sides 2, 3, and 6 fail because 2 + 3 = 5, which is not strictly greater than 6, making a closed triangle geometrically impossible.

Calculator Modes Explained

This triangle inequality calculator supports three distinct computation modes:

Validity Check: Enter all three sides a, b, and c to determine immediately whether they satisfy all three triangle inequality conditions.
Third-Side Range: Enter two known sides to compute the open interval of valid values for the missing third side using the formula |a − b| < c < a + b.
Comparison Values: Enter all three sides to display all pairwise sums and differences for direct numerical inspection and manual verification.

Real-World and Academic Applications

The triangle inequality appears throughout mathematics, physics, and engineering. In metric spaces, it serves as a defining axiom: the distance from point A to point C cannot exceed the distance from A to B plus the distance from B to C. This principle underlies GPS triangulation, network routing optimization, and computer graphics collision detection. In structural engineering, the theorem ensures that triangular braces and trusses close geometrically, preventing physically impossible configurations. Robotics path planning and machine learning nearest-neighbor search algorithms also rely on triangle inequality reasoning to prune candidate distances efficiently.

Special Cases and Edge Conditions

An equilateral triangle (a = b = c = s) trivially satisfies all conditions since s + s = 2s > s for any positive s. An isosceles triangle requires checking only the inequalities involving the unequal side. A degenerate triangle arises when one side equals exactly the sum of the other two — this lies on the strict boundary and does not qualify as a valid triangle. The classic right triangle with legs 3 and 4 and hypotenuse 5 satisfies 3 + 4 = 7 > 5, 3 + 5 = 8 > 4, and 4 + 5 = 9 > 3, confirming full validity.

Authoritative Sources

The formal secondary-level treatment of the triangle inequality is outlined in the Wisconsin Department of Public Instruction Performance Level Descriptors for Mathematics, which benchmarks student mastery of the theorem. Advanced generalizations linking the classical theorem to analytic inequalities are examined in Ceva's Triangle Inequalities from Western Washington University. Additional worked proofs and exam-level applications appear in the Rutgers University Math 250 Final Exam Review, and complex-analysis extensions are covered in the UNCW Complex Comprehensive Review.

Reference

Frequently asked questions

What is the triangle inequality theorem and why does it matter?

The triangle inequality theorem states that the sum of any two sides of a triangle must be strictly greater than the third side. For sides a, b, and c, all three conditions must hold simultaneously: a + b > c, a + c > b, and b + c > a. If any single condition fails, the three lengths cannot enclose a finite area and therefore cannot form a valid triangle.

How do you find the range of a missing third side using the triangle inequality?

Given two known sides a and b, the missing third side c must satisfy the compound inequality |a - b| < c < a + b. For example, if a = 6 and b = 9, then |6 - 9| = 3 and 6 + 9 = 15, so c must lie strictly between 3 and 15. Both endpoints are excluded because c = 3 or c = 15 would produce a degenerate, collinear configuration rather than a true closed triangle.

Can a triangle have sides of length 3, 4, and 8?

No. Sides 3, 4, and 8 violate the triangle inequality. The critical condition 3 + 4 = 7 must be strictly greater than 8, but 7 is less than 8, so the condition fails. Although the other two conditions hold (3 + 8 = 11 > 4 and 4 + 8 = 12 > 3), all three inequalities must be satisfied simultaneously, so these three lengths cannot form a valid triangle.

Does the triangle inequality apply to right triangles and obtuse triangles?

Yes, the triangle inequality applies universally to all triangle types: right, obtuse, acute, equilateral, and isosceles. The classic 3-4-5 right triangle satisfies 3 + 4 = 7 > 5, confirming validity. An obtuse triangle with sides 5, 6, and 9 satisfies 5 + 6 = 11 > 9, 5 + 9 = 14 > 6, and 6 + 9 = 15 > 5, so it is fully valid even though its largest angle exceeds 90 degrees.

What is the difference between the triangle inequality and the reverse triangle inequality?

The standard triangle inequality states that any side length is bounded by the sum of the other two: |a - b| < c < a + b, providing both a lower and upper bound. The reverse triangle inequality, used in real analysis and metric space theory, states that the absolute difference of two distances is at most the third: |d(A,C) - d(B,C)| is less than or equal to d(A,B). The reverse form isolates only the lower bound and is critical in convergence proofs and norm estimations in functional analysis.

How is the triangle inequality applied outside of basic geometry?

The triangle inequality is a foundational axiom in metric space theory, requiring that the distance between any two points never exceeds the sum of distances through any intermediate point. Practical applications include GPS triangulation, network shortest-path routing algorithms, machine learning nearest-neighbor searches that prune candidates using distance bounds, and computer graphics collision detection. In linear algebra, the vector form asserts that the norm satisfies ||u + v|| is less than or equal to ||u|| + ||v||, which is essential in signal processing, optimization, and numerical analysis.