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Geometry · Triangles

General Triangle Solver — SSS, SAS, ASA & SSA

Solve a triangle from SSS, SAS, ASA/AAS, or SSA data, including both valid solutions in the ambiguous SSA case.

Formula shown Runs locally Reviewed Aug 11, 2026
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Equation guide

Triangle cases use S = side and A = angle

Side-angle pairing

a ↔ A, b ↔ B, c ↔ CA+B+C=180°

Law of cosines

a²=b²+c²−2bc cos A

Law of sines

a/sin A=b/sin B=c/sin CSSA may have 0, 1, or 2 solutions
Geometry · Triangle cases

Solve every determined triangle case

Case notation: S = side and A = angle. SSS means three sides; SAS means two sides and their included angle; ASA means two angles and their included side; SSA means two sides and a non-included angle. Lowercase sides a, b, c are opposite uppercase angles A, B, C. ASA also covers AAS after relabeling.

Inputs and displayed decimal results use at most three decimal places. Exact internal values are retained during calculation.

K = √[s(s−a)(s−b)(s−c)]

Enter the three values required by the selected case.

ABC

Using the model

Symbols, assumptions and limitations

Interpretation notes

Lowercase sides a, b, and c lie opposite uppercase angles A, B, and C. SSS and SAS use the law of cosines; ASA/AAS and SSA use the law of sines. SSA may produce zero, one, or two triangles, and every candidate is checked for positive angles and strict triangle inequality.

Worked examples

See the method in practice

01

3–4–5 SSS triangle

Sides 3, 4, and 5 give a 90° angle opposite side 5, area 6, and perimeter 12.

02

Ambiguous SSA data

Side a = 10, angle A = 30°, and side b = 12 produce two possible values of angle B and therefore two distinct triangles.

Questions

Frequently asked

Why can SSA have two answers?

The sine of an acute angle equals the sine of its supplementary obtuse angle. Both may fit before the remaining-angle check.

What does included angle mean in SAS?

It is the angle between the two entered sides, so it directly determines the opposite third side through the law of cosines.

How does ASA cover AAS?

Two known angles determine the third angle. Relabeling the known side then reduces either arrangement to the same law-of-sines calculation.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax Precalculus — Non-right Triangles

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology