Formula and Method for Solving Oblique Triangles
An oblique triangle is any triangle that does not have a 90° angle — it can be acute (every angle under 90°) or obtuse (one angle over 90°). Because there is no right angle to anchor SOHCAHTOA, solving an oblique triangle (finding all three sides and all three angles) requires the Law of Sines and the Law of Cosines. This calculator supports the five standard ways a triangle can be specified — SSS, SAS, ASA, AAS, and the ambiguous SSA case — using the usual convention that side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.
How the calculation works
Pick the "Given information" that matches what you know, then fill in only those fields (the rest are ignored for that case):
- SSS (three sides): the Law of Cosines finds each angle, e.g. cos A = (b² + c² − a²) / (2bc); the third angle follows from A + B + C = 180°. Area comes from Heron's formula.
- SAS (two sides + included angle): the Law of Cosines finds the third side, c² = a² + b² − 2ab·cos C, then the Law of Cosines again finds a second angle and subtraction gives the third.
- ASA (two angles + included side) and AAS (two angles + a non-included side): the missing angle is 180° minus the other two, then the Law of Sines, a/sin A = b/sin B = c/sin C, finds the remaining sides.
- SSA (two sides + a non-included angle): this is the "ambiguous case." The Law of Sines can produce zero, one, or two valid triangles depending on whether the given side opposite the known angle is long enough (and how it compares to the other side) to close the triangle.
Common mistakes
- Included vs. non-included angle: SAS needs the angle sandwiched between the two given sides. Entering an angle that is not between the two sides turns the problem into the ambiguous SSA case instead.
- Mismatched opposite pairs: keep the a↔A, b↔B, c↔C pairing consistent — swapping which side is "a" versus "b" changes which angle you are solving for.
- Ignoring the SSA ambiguous case: two sides and a non-included angle can match two different triangles, one, or none at all. Don't assume the first answer you compute is the only one.
- Triangle inequality violations: in SSS mode, each side must be shorter than the sum of the other two, or no triangle can close.
Real-world applications
- Surveying and land measurement use triangulation, which is fundamentally solving oblique triangles from measured angles and a known baseline distance.
- Navigation and aviation compute bearings, headings, and distances between points that rarely form a convenient right angle.
- Structural engineering and construction size trusses, braces, and roof pitches that meet at oblique angles.
- Astronomy uses oblique triangle solutions in parallax calculations to estimate distances to nearby stars.
Frequently Asked Questions
What is an oblique triangle?
An oblique triangle is any triangle that does not contain a 90° angle — it can be acute (all angles less than 90°) or obtuse (one angle greater than 90°). Because there is no right angle, you solve it with the Law of Sines and Law of Cosines instead of basic right-triangle trigonometry.
Which law should I use — Law of Sines or Law of Cosines?
Use the Law of Sines when you know two angles and a side (ASA or AAS) or two sides and a non-included angle (SSA). Use the Law of Cosines when you know three sides (SSS) or two sides and the included angle between them (SAS).
What is the ambiguous SSA case?
When you know two sides and an angle opposite one of them, that information can match zero, one, or two different triangles depending on how the sides and angle compare. This calculator checks all outcomes and reports both triangles when two valid solutions exist.
How do I find the area of an oblique triangle?
If you know two sides and the included angle, Area = ½ab sin C. If you know all three sides, use Heron's formula: s = (a+b+c)/2 and Area = √(s(s−a)(s−b)(s−c)).