Triangle Inequality Checker & Classifier

About the Triangle Inequality Checker & Classifier

The triangle inequality theorem is one of the most fundamental results in geometry: for any three lengths to form a triangle, the sum of every pair must be strictly greater than the third. In symbols: a + b > c, a + c > b, and b + c > a. If any one of these fails, no triangle can be constructed.

This calculator checks all three inequalities simultaneously and reports the surplus (how much the sum exceeds the third side) or the deficit (how much more length is needed). When the sides are valid, it goes further — classifying the triangle by its angles (acute if a² + b² > c² for all orderings, right if equality holds, obtuse otherwise) and by its sides (equilateral, isosceles, or scalene).

For valid triangles, the calculator also computes the area (Heron's formula), all three angles (law of cosines), perimeter, circumradius (R), and inradius (r). For degenerate cases where a + b = c exactly (the three points are collinear), it warns that the "triangle" has zero area. For invalid cases, it shows exactly which inequality fails and by how much, helping you understand what would need to change.

The triangle inequality appears throughout mathematics and its applications — from metric spaces in analysis to network routing in computer science. In construction and engineering, checking whether three members can form a rigid triangle is a basic feasibility test. This calculator makes that check instant.

When This Page Helps

This calculator is useful when you need more than a yes-or-no triangle check. It shows each inequality separately, measures the surplus or deficit, and then classifies the triangle only if the side set is actually valid. That makes it useful for classroom demonstrations, quick feasibility checks in design work, and debugging geometry problems where one side length might have been copied incorrectly.

How to Use the Inputs

1. Enter the three candidate side lengths a, b, and c.
2. Choose a measurement unit.
3. Click a preset to try known valid or invalid sets.
4. View the inequality check table with surplus/deficit for each pair.
5. See the verdict: valid triangle, degenerate, or invalid.
6. If valid, explore the full classification, area, angles, and radii.
7. If invalid, see how much more length is needed.

Example Calculation

Tips & Best Practices

• The triangle inequality generalizes to all metric spaces — it is the defining property of a "distance" function.
• A degenerate triangle (a+b = c exactly) has zero area and represents three collinear points.
• The largest angle is always opposite the longest side — useful for quick classification.
• To test for a right triangle, sort the sides and check if the sum of the two smaller squares equals the largest square.
• If an inequality fails, the deficit tells you the minimum extra length needed on the shorter sides.

Why the Inequalities Must Be Strict

A triangle only exists when every pairwise sum is strictly greater than the remaining side. Equality is not enough, because it collapses the figure into a straight segment with zero area. That strict comparison is why a set such as 3, 4, and 7 is not a thin triangle but a degenerate one, and why any smaller sum fails completely.

Reading Surplus and Deficit Values

The most practical part of an inequality check is often not the pass or fail label but the margin. A positive surplus tells you how much room there is before the triangle would collapse, while a negative value tells you exactly how far the data is from being valid. This is useful when checking measurements from field work, CAD sketches, or hand calculations where rounding or transcription may have introduced an error.

Classifying Only After Validity

Angle and side classification make sense only after the triangle inequality has passed. Once the figure is valid, the same side set can be analyzed further as acute, right, or obtuse and as equilateral, isosceles, or scalene. Keeping those stages separate is good mathematical hygiene: first prove the triangle can exist, then describe what kind of triangle it is.

Frequently Asked Questions

• What is the triangle inequality theorem?

  It states that for any triangle with sides a, b, c: the sum of any two sides must be greater than the third side. All three conditions must hold: a+b>c, a+c>b, b+c>a.

• What happens when a + b = c exactly?

  The three points are collinear (on a straight line). This is called a degenerate triangle — it has zero area and no interior angles.

• How do I determine if a triangle is acute, right, or obtuse?

  Sort sides so a ≤ b ≤ c. Compute a² + b² vs c². If a²+b² > c²: acute. If equal: right. If less: obtuse.

• What is a scalene triangle?

  A triangle where all three sides have different lengths (and therefore all three angles are different).

• Can a triangle have sides 1, 1, and 3?

  No. 1 + 1 = 2, which is not greater than 3. The triangle inequality fails.

• Why is the triangle inequality important in real life?

  In construction, it determines whether three beams can form a rigid frame. In navigation, it constrains shortest-path distances. In computer science, it underpins metric-based search algorithms.