Understanding the SSS Triangle Problem
An SSS (Side-Side-Side) triangle is fully determined the moment you know all three side lengths. Unlike other scenarios where you might have angles or a mix of angles and sides, SSS gives you complete information—no ambiguity, no multiple solutions.
Once you have sides a, b, and c, you can calculate:
- All three interior angles using the law of cosines
- The total area using Heron's formula or the sine rule
- The perimeter (trivially: a + b + c)
This makes SSS one of the most straightforward triangle configurations to work with, and it's the foundation of the SSS congruence criterion: two triangles with identical side lengths are always congruent.
Formulas for SSS Triangles
The core approach relies on the law of cosines, which relates the sides of any triangle to its angles. Once angles are known, Heron's formula provides the area without needing any height measurement.
cos(α) = (b² + c² − a²) ÷ (2bc)
cos(β) = (a² + c² − b²) ÷ (2ac)
γ = 180° − α − β
s = (a + b + c) ÷ 2
Area = √[s(s − a)(s − b)(s − c)]
a, b, c— The lengths of the three sides of the triangleα, β, γ— The interior angles opposite sides a, b, and c respectivelys— The semi-perimeter, equal to half the perimeterArea— The total area enclosed by the three sides
How to Use This Calculator
Enter the three side lengths in any order (the calculator will identify which is which). Within moments, you'll receive all three angles in degrees and the area in your chosen unit.
The calculator also reports the perimeter automatically. If you already know one or more angles and want to verify your sides are correct, you can input angles instead and solve backwards—the tool supports bidirectional calculation.
All results update live, so experimenting with different side lengths is instant and risk-free. No need to manually apply the law of cosines or expand Heron's formula by hand.
Common Pitfalls and Practical Tips
Keep these points in mind when working with SSS triangles to avoid errors and get the most from your calculations.
- Triangle Inequality Constraint — The sum of any two sides must exceed the third side. If you enter sides 2, 3, and 5, they won't form a valid triangle because 2 + 3 = 5 (equality, not strict inequality). The calculator will flag this; make sure your measurements don't violate this rule.
- Precision in Angle Rounding — When the law of cosines produces angles, rounding each independently and then subtracting can introduce error. This calculator computes the third angle as 180° minus the sum of the first two, preserving accuracy. If you compute angles manually, sum the first two to maximum precision before subtracting.
- Heron's Formula Stability — For very flat triangles (where one angle approaches 180°), Heron's formula can suffer from numerical precision loss. The calculator uses a numerically stable variant. If you implement this yourself, consider alternative area formulas or higher-precision arithmetic for extreme geometries.
- Unit Consistency — Ensure all three side lengths are in the same unit (metres, feet, inches, etc.). The area will be in the square of that unit. If your inputs mix units, convert them first; the calculator assumes homogeneity.
SSS Congruence and Real-World Applications
Two triangles are congruent (identical in shape and size) if their corresponding sides are equal. This SSS congruence criterion is one of the most useful in geometry and appears throughout engineering, surveying, and architecture.
In practice, if you measure three sides of a physical triangle—say, the frame of a roof truss or a surveyed plot of land—you've captured all the information needed to reconstruct it exactly elsewhere. No matter how you orient or flip the triangle, its three side lengths uniquely determine angles and area.
This is why SSS triangles are so common in real-world problems: you often measure distances directly, whereas angles may be harder to gauge accurately.