Triangle Calculator

A triangle has 6 properties: 3 sides and 3 angles. You need at least 3 known values (with at least one being a side) to solve for all 6. Select the combination you know:

• SSS - all three sides known. Uses the Law of Cosines: cos(A) = (b² + c² - a²) / (2bc) to find angles.
• SAS - two sides and the included angle between them. Uses Law of Cosines to find the third side, then Law of Sines for remaining angles.
• ASA - two angles and the side between them. The third angle = 180° - A - B. Uses Law of Sines for the remaining sides.
• AAS - two angles and a non-included side. Find the third angle first, then use Law of Sines.

Note: SSA (two sides and a non-included angle) can have 0, 1, or 2 valid triangles depending on the values, known as the "ambiguous case."

Angles sum:    A + B + C = 180°

Law of Sines:  a/sin(A) = b/sin(B) = c/sin(C)

Law of Cosines:
a² = b² + c² - 2bc·cos(A)
b² = a² + c² - 2ac·cos(B)
c² = a² + b² - 2ab·cos(C)

Area formulas:
Area = (1/2) × base × height
Area = (1/2) × a × b × sin(C)
Area = √(s(s-a)(s-b)(s-c))  [Heron's]
where s = (a+b+c)/2

Worked example (SAS): sides a = 8, b = 11, angle C = 42°