Enter known sides and angles to solve any non-right triangle using Law of Sines and Cosines
Side a
Side b
Side c
Angle A (deg)
Angle B (deg)
Angle C (deg)
Enter 3 known values (SSS, SAS, ASA, AAS, or SSA). Leave unknowns empty.
Result
a
-
b
-
c
-
Angle A
-
Angle B
-
Angle C
-
Detailed Derivation
Law of Sines & Law of Cosines
Law of Sines: a/sinA = b/sinB = c/sinC = 2R
Law of Cosines: c^2 = a^2 + b^2 - 2ab cosC
Area: K = (1/2)ab sinC
Angle Sum: A + B + C = 180
Oblique triangles are solved using the Law of Sines and Law of Cosines together with the angle sum property. The method used depends on which values are known.
⚠Enter at least 3 known values. For SSS, enter all three sides. For SAS, enter two sides and the included angle.
What Is an Oblique Triangle?
An oblique triangle has no right angle. It requires different solving methods than right triangles. The Law of Sines and Law of Cosines are the primary tools for finding all unknown sides and angles.
Law of Sines
a/sinA = b/sinB = c/sinC. Best for ASA, AAS, and SSA cases. The common ratio equals the circumdiameter (2R).
Law of Cosines
c^2 = a^2 + b^2 - 2ab cosC. Generalizes the Pythagorean theorem. Use for SSS and SAS cases.
Ambiguous Case
SSA can produce 0, 1, or 2 triangles. Check the height h = b sinA. The tool detects this automatically.
Area Formulas
K = (1/2)ab sinC or Heron formula. All formulas give the same result if calculations are correct.
Teaching Example: SSS case: a=5, b=6, c=7. Using Law of Cosines: cosA = (b^2+c^2-a^2)/(2bc) = (36+49-25)/(84) = 0.714. A = 44.4 deg. Similarly B = 57.1 deg, C = 78.5 deg. Sum = 180 deg. Area = (1/2)(5)(6)sin(57.1) = 14.7.
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