Triangle Solver
SSS Mode: enter all three side lengths to solve for all three angles.
From Three Sides to a Fully Solved Triangle
Give this calculator the three side lengths of any valid triangle, and it hands back every angle inside it. There's no need to measure a single angle directly — the geometry of the triangle already determines them, and the law of cosines extracts each one in turn.
The Formula
Applying the law of cosines once per angle, then closing out with the angle sum:
B = acos((a²+c²−b²)/(2ac))
C = 180° − A − B
Before solving, the three lengths are checked against the triangle inequality — the sum of any two sides must exceed the third, or no triangle exists.
Worked Example
For a triangle with sides a = 5, b = 6, c = 7:
| Angle | Opposite Side | Value |
|---|---|---|
| A | a = 5 | 44.4153° |
| B | b = 6 | 57.1217° |
| C | c = 7 | 78.4630° |
The three angles sum to exactly 180°, as they must in any planar triangle — a useful sanity check on the result.
Where This Matters
- Structural analysis — a truss's member lengths are fixed by design, and the interior angles determine how load transfers through each joint.
- CNC and CAD verification — confirming a triangular cutout or panel matches its intended angles from measured edge lengths.
- Land surveying — reconstructing a plot's interior angles from chained (taped) side measurements alone.
How to Use This Calculator
- Enter Side a, Side b, and Side c.
- Select Calculate.
- Read off angles A, B, and C, each opposite its matching side.
Related Calculations
Only have some of the sides and angles? The Sine Rule Calculator and Cosine Rule Calculator handle partial information.