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Solve any triangle from SSS, SAS, SSA, AAS, or ASA. Outputs all sides, angles, area, perimeter, and heights.
Key Triangle Formulas:
Law of Cosines: c² = a² + b² - 2ab cos(C)
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
Heron's Area: s=(a+b+c)/2; Area = sqrt(s(s-a)(s-b)(s-c))
Height from side a: h_a = 2 × Area / a
These are triangle congruence conditions. SSS = three sides known. SAS = two sides and included angle. SSA = two sides and a non-included angle (may give two solutions). AAS = two angles and a non-included side. ASA = two angles and the included side.
The Law of Cosines generalizes the Pythagorean theorem: c² = a² + b² - 2ab cos(C). It allows you to find a missing side when you know two sides and the angle between them, or to find angles when you know all three sides.
The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C). It relates the sides of a triangle to the sines of their opposite angles, useful when you know a side and its opposite angle.
In the SSA (ambiguous) case, the given angle may produce two valid triangles, one triangle, or no triangle depending on the side lengths. The calculator returns one valid solution when possible.
The calculator uses Heron's formula: compute the semi-perimeter s = (a+b+c)/2, then Area = sqrt(s(s-a)(s-b)(s-c)). Heights are derived as h = 2 × Area / base.