Triangle Side-Angle Calculator (Triangle Solver)

About the Triangle Side-Angle Calculator (Triangle Solver)

The **Triangle Side-Angle Calculator** is a comprehensive triangle solver. Given any valid combination of three known elements — sides and angles — it determines all remaining unknowns and reports the full set of triangle properties.

Five classic input combos are supported:

- **SSS** — three sides known. All angles can be found via the Law of Cosines. - **SAS** — two sides and the included angle. The third side comes from the Law of Cosines, then the remaining angles from the Law of Sines. - **ASA** — two angles and the included side. The third angle is 180° minus the other two, and the remaining sides follow from the Law of Sines. - **AAS** — two angles and a non-included side. Equivalent to ASA after computing the third angle. - **SSA** — two sides and a non-included angle (the ambiguous case). There may be zero, one, or two valid triangles.

Beyond just sides and angles, the calculator outputs the area, perimeter, inradius, circumradius, altitude from each vertex, and the type of triangle (acute, right, or obtuse). Visual bars show side and angle proportions at a glance, and a reference table explains when to use each input combination.

Whether you are a geometry student, a surveyor, an architect, or an engineer, the page replaces tedious hand work with the computed triangle values laid out clearly.

When This Page Helps

A general triangle solver is useful because real problems do not arrive in one neat format. One question gives you three sides, another gives two angles and a side, and the next introduces the ambiguous SSA case where there may be two valid answers. This calculator keeps those combinations separate and solves each one using the correct sequence of cosine and sine relationships.

It is especially helpful for checking whether a problem has a unique triangle or multiple solutions. Seeing the full solved triangle, along with area and radius values, makes it much easier to understand what a given side-angle combination actually determines.

How to Use the Inputs

1. Select the input combination that matches the information you have (SSS, SAS, ASA, AAS, or SSA).
2. Enter the known values — sides in any consistent unit, angles in degrees.
3. Click a preset button to load a classic example for instant exploration.
4. Review the output cards for all solved sides, angles, area, and perimeter.
5. Check the visual bars for a proportional view of the triangle.
6. Consult the reference table for an overview of all input combinations and the solving strategy for each.

Example Calculation

Tips & Best Practices

• SSA can produce 0, 1, or 2 triangles — this calculator handles all three outcomes.
• If the sum of two angles ≥ 180°, no triangle exists — check your input.
• For right triangles, SAS with C = 90° reduces to the Pythagorean theorem.
• The circumradius R relates to side and angle: R = a / (2 sin A).
• Enter angles in degrees; the calculator converts to radians internally.

Different Data Sets Solve Triangles in Different Ways

A triangle can be determined by several combinations of sides and angles, but each combination has its own logic. SSS starts with shape from side lengths, SAS uses an included angle to lock in the third side, and ASA or AAS use the angle sum to unlock the missing side lengths through the Law of Sines. Treating all of those as the same kind of exercise hides the reasoning that makes triangle solving work.

The SSA Ambiguous Case Is Worth Studying

SSA is the unusual case because the same data can describe zero, one, or two triangles. That happens because the Law of Sines determines a sine value, and the same sine can correspond to two different angles between 0° and 180°. If both resulting geometries satisfy the triangle conditions, both solutions are valid.

How to Read a Solver Output Well

After a triangle is solved, do not stop at the missing numbers. Compare the side lengths to the opposite angles, check whether the triangle is acute, right, or obtuse, and see whether the perimeter and area agree with the overall scale you expected. A good solver should support interpretation, not just produce a list of values.

Frequently Asked Questions

• What is the ambiguous case (SSA)?

  When you know two sides and an angle opposite one of them, the Law of Sines can yield 0, 1, or 2 valid triangles depending on relative sizes.

• Can I solve a triangle with only three angles?

  No. Three angles fix the shape but not the size. You need at least one side to determine a unique triangle.

• What is the inradius?

  The inradius r is the radius of the largest circle that fits inside the triangle. r = Area / s, where s is the semi-perimeter.

• What is the circumradius?

  The circumradius R is the radius of the circle passing through all three vertices. R = a / (2 sin A).

• How do I know which combination I have?

  Count your known sides (S) and angles (A). Common combos are SSS (3 sides), SAS (2 sides + included angle), ASA (2 angles + included side), AAS (2 angles + non-included side), and SSA (2 sides + non-included angle).

• Are the results exact?

  Computations use double-precision floating point (about 15 significant digits). Results are displayed rounded to your chosen precision.