Cosine Rule
a² = b² + c² − 2bc cos A
Use SSS to recover an angle or SAS to calculate the side opposite the included angle. Unlike Pythagoras, this works for acute and obtuse triangles.
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GEOMETRY / TRIGONOMETRY
Non-Right-Angled Triangle Solver
Find missing sides, angles, area and perimeter using the Sine Rule and Cosine Rule.
01
Choose what you know
Angles are in degrees. Side a is opposite angle A, b is opposite B, and c is opposite C.
The SSA case is checked for zero, one, or two valid triangles. A 90° triangle is accepted as a special case, never assumed.
02
Understanding the mathematics
Sine Rule
a / sin A = b / sin B = c / sin C
Relates each side to its opposite angle. In SSA, also test the supplementary angle rather than trusting inverse sine alone.
Area of a triangle
K = ½ bc sin A
K = √[s(s − a)(s − b)(s − c)]
The first formula uses two sides and their included angle. Heron’s formula uses all three sides, with s = (a + b + c) / 2.
The ambiguous SSA case
sin B = b sin A / a
An SSA input may have no triangle, exactly one triangle, or two distinct triangles. Every valid possibility is shown separately.
