Oblique Triangle Solver (Law of Sines & Cosines)
Solve any non-right triangle given two angles and a side (AAS/ASA), two sides and an angle (SAS/SSA), or find its area. Ideal for surveying, navigation, and geometry problems.
About this calculator
An oblique triangle has no right angle, so the Pythagorean theorem does not apply directly. Two laws govern its solution. The Law of Sines states: a/sin A = b/sin B = c/sin C, used for AAS and ASA cases. The Law of Cosines states: c² = a² + b² − 2ab·cos C, used for SAS cases to find the missing side. Area under SAS is: Area = ½·a·b·sin C. The ambiguous SSA case uses the Law of Sines to find a missing angle: sin B = b·sin A / a, which may yield zero, one, or two valid triangles. Selecting the correct case in this calculator ensures the appropriate formula is applied automatically.
How to use
SAS example: side a = 7, side b = 10, included angle C = 60°. The Law of Cosines gives: c = √(7² + 10² − 2·7·10·cos 60°) = √(49 + 100 − 140·0.5) = √(149 − 70) = √79 ≈ 8.888 units. Area = ½·7·10·sin(60°) = 35·0.8660 ≈ 30.31 square units. Enter knownSide1 = 7, knownAngle1 = 60, knownValue3 = 10, select case 'SAS', and choose 'missing side' or 'area' to get each result.
Frequently asked questions
What is the difference between the Law of Sines and the Law of Cosines?
The Law of Sines relates each side to the sine of its opposite angle and is most efficient when you know two angles and a side (AAS or ASA). The Law of Cosines relates all three sides and one angle, making it necessary when two sides and the included angle are known (SAS) or all three sides are known (SSS). Both laws together allow any triangle to be fully solved given sufficient information. The Law of Cosines reduces to the Pythagorean theorem when the angle is 90°.
Why does the SSA case sometimes produce two different triangles?
In the SSA (side-side-angle) configuration, knowing two sides and a non-included angle does not uniquely determine a triangle. The given side opposite the known angle may 'swing' to two different positions, producing two valid triangles with different shapes. This is called the ambiguous case. It occurs when the opposite side is shorter than the adjacent side but longer than the height of the triangle. The calculator returns the principal value; you should check whether a second solution exists by considering the supplementary angle.
How do I find the area of a triangle when only two sides and the included angle are known?
Use the SAS area formula: Area = ½·a·b·sin C, where a and b are the two known sides and C is the angle between them. This formula works because ½·base·height can be rewritten using the sine of the included angle to express the height in terms of the second side. For example, sides of 5 and 8 with an included angle of 50° give Area = ½·5·8·sin(50°) ≈ 15.32 square units. This approach avoids the need to find the third side first.