Triangle Inequality Theorem Calculator
Verify if three side lengths can form a valid triangle using the Triangle Inequality Theorem. Get instant results with triangle classification and area.
Triangle Inequality Theorem Calculator
Calculator
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Formula: a + b > c, a + c > b, b + c > a
Worked example — Valid triangle | Scalene, Obtuse | Perimeter = 22 | Area = 16.2481
Formula
a + b > c, a + c > b, b + c > a
For three lengths a, b, and c to form a valid triangle, the sum of any two sides must be strictly greater than the third side. All three conditions must hold simultaneously.
Worked Examples
Example 1: Valid Triangle Check
Problem:Determine if sides 5, 7, and 10 can form a valid triangle using the Triangle Inequality Theorem.
Solution:Check 1: 5 + 7 = 12 > 10 (Pass) Check 2: 5 + 10 = 15 > 7 (Pass) Check 3: 7 + 10 = 17 > 5 (Pass) All three inequalities are satisfied. Perimeter = 5 + 7 + 10 = 22 Semi-perimeter s = 11 Area = sqrt(11 * 6 * 4 * 1) = sqrt(264) = 16.2481
Result:Valid triangle | Scalene, Obtuse | Perimeter = 22 | Area = 16.2481
Example 2: Invalid Triangle Detection
Problem:Can sides 1, 2, and 5 form a valid triangle?
Solution:Check 1: 1 + 2 = 3 > 5? No, 3 is NOT greater than 5 (Fail) Check 2: 1 + 5 = 6 > 2 (Pass) Check 3: 2 + 5 = 7 > 1 (Pass) Since the first inequality fails, these lengths cannot form a triangle.
Result:Invalid triangle | 1 + 2 = 3 which is less than 5
Frequently Asked Questions
What is the Triangle Inequality Theorem and why is it important?
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining third side. This must hold true for all three possible combinations of sides. It is one of the most fundamental theorems in Euclidean geometry and serves as the necessary and sufficient condition for three positive lengths to form a valid triangle. Without this theorem, we would have no way to quickly verify whether a given set of three measurements can actually be constructed into a closed triangular shape in a flat plane.
How do you check all three conditions of the Triangle Inequality Theorem?
To fully verify the theorem, you must test three separate inequalities for sides a, b, and c. First, check that a + b > c. Second, check that a + c > b. Third, check that b + c > a. All three conditions must be satisfied simultaneously for the three lengths to form a valid triangle. If even one condition fails, the segments cannot connect to form a closed triangle. In practice, you only need to verify that the sum of the two smallest sides exceeds the largest side, because the other two inequalities are automatically satisfied when that condition holds true.
What happens when the sum of two sides equals the third side exactly?
When the sum of two sides equals the third side exactly (for example, a + b = c), the three segments form a degenerate triangle, which is essentially a straight line with zero area. This is not considered a valid triangle in standard Euclidean geometry because the three points become collinear rather than forming a two-dimensional shape. The strict inequality (greater than, not greater than or equal to) is required precisely to exclude this degenerate case. In computational geometry, a small epsilon tolerance is often used to handle floating-point precision issues when testing for degeneracy.
How does the Triangle Inequality Theorem relate to the shortest path between two points?
The Triangle Inequality Theorem is closely connected to the fundamental geometric principle that a straight line is the shortest distance between two points. If you travel from point A to point C directly, the distance is always less than or equal to traveling from A to B and then B to C. This means AC is always less than or equal to AB + BC, which is exactly the triangle inequality. This concept extends far beyond basic geometry into metric spaces, graph theory, and optimization algorithms. The triangle inequality is an axiom of every valid distance metric used in mathematics and computer science.
Can the Triangle Inequality Theorem be applied to non-Euclidean geometry or higher dimensions?
Yes, the triangle inequality generalizes broadly across mathematics. In spherical geometry, the inequality still holds for geodesic distances on the surface of a sphere, although the strict form may differ due to curvature. In hyperbolic geometry, the inequality applies with adjustments for negative curvature effects. In higher-dimensional Euclidean spaces, the triangle inequality extends naturally to distances between points in three dimensions and beyond. In abstract mathematics, any metric space must satisfy the triangle inequality as one of its defining axioms, making it essential in functional analysis, topology, and many branches of applied mathematics.
What range of values can the third side take when two sides are known?
Combining the three inequalities gives a two-sided bound: the third side c must satisfy the absolute value of (a minus b) is less than c, and c is less than a plus b. With a equal to 5 and b equal to 7, the third side must lie strictly between 2 and 12. Both bounds are strict, since hitting either endpoint collapses the figure into a straight segment. This is why the theorem is sometimes stated as the third side lying between the difference and the sum of the other two.
Why does Heron's formula fail when the inequality is violated?
Heron's formula gives the area as the square root of s(s - a)(s - b)(s - c), where s is the semiperimeter (a + b + c) divided by 2. If a set of lengths breaks the inequality, the longest side exceeds the sum of the other two, so s minus that side is negative while the other factors stay positive. The product under the radical becomes negative and no real area exists. When the sum exactly equals the third side, one factor is 0, giving zero area, which is the degenerate case.
How do the side lengths tell you whether the triangle is acute, right, or obtuse?
Take the longest side c and compare c squared with a squared plus b squared, which is the law of cosines with the cosine term isolated. If a squared plus b squared is greater than c squared the largest angle is acute; if the two are equal the triangle is right-angled and the Pythagorean theorem holds exactly; if the sum is smaller the largest angle is obtuse. For sides 5, 7 and 10, the sum 25 plus 49 equals 74, which is less than 100, so the triangle is obtuse.
Why is the perimeter of any triangle always more than twice its longest side?
Let c be the longest side. The inequality a plus b is greater than c can be added to c on both sides, giving a plus b plus c greater than 2c, so the perimeter always exceeds double the longest side. Equivalently, no single side can be as much as half the perimeter. This provides a fast one-line screen on a set of measurements: if any length reaches or exceeds half the total, the three lengths cannot close into a triangle.
If three lengths pass the test, can they build more than one differently shaped triangle?
No. Once the inequality is satisfied, the three lengths determine a single triangle up to congruence, which is the side-side-side congruence criterion. All three angles are then fixed and can be recovered from the law of cosines, so the area, the circumradius and the inradius are all determined as well. Reflecting or rotating the figure produces a congruent copy, not a genuinely different shape, which is why side lengths alone are enough to specify a triangle but not a quadrilateral.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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