Sine hides a circle
The Law of Sines really says each side equals 2R·sin(angle), where R is the circumcircle radius—your triangle is inscribed in a single invisible circle.
Circumcircle secret
Trigonometric Triangle Solver — AAS, ASA, SAS, SSS, SSA
Diagram & Inputs
2 dp
Tip: Any three values with ≥1 side. Convention: side a is opposite angle A, etc.
Results
Steps
How This Trigonometric Solver Works
Enter any three values with at least one side (SSS, SAS, ASA, AAS, or SSA). The solver auto-detects the case, applies the Law of Cosines and/or Law of Sines, and shows the algebra steps. For SSA, the tool checks whether there are 0, 1, or 2 valid solutions and displays each one.
- Law of Cosines:
a² = b² + c² − 2bc·cos(A) - Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C) - Area (Heron, SSS):
s=(a+b+c)/2,Area = √(s(s−a)(s−b)(s−c)) - Area (two sides + included angle):
Area = ½ab·sin(C)(and cyclic) - Angle sum:
A + B + C = 180°
All computation is performed privately in your browser.
5 Fun Facts about Trig & Triangles
Area peaks at 90°
For fixed sides b and c, Area = ½bc·sin(A). Because sine maxes at 1, the largest possible area happens when A is exactly 90°.
Max-area tip
Ambiguous twins
SSA can spawn two valid angles because sin(θ) = sin(180°−θ). One triangle sits tall, the other is its squat mirror—until the “mirror” breaks when the height exceeds the long side.
Double-check SSA
Units can fake impossibles
Plugging radians into a degree field (or vice versa) can make a perfectly valid triangle look impossible. A tiny “1” radian is ~57.3°—always match the toggle to your inputs.
Unit sanity
Longest side tells the boss
The largest angle always faces the longest side. If your computed angles and side ordering disagree, something’s off—an easy sniff test before trusting the numbers.
Quick sanity check
Triangle Solver: FAQs
Which inputs are valid?
Any three values with at least one side (SSS, SAS, ASA, AAS, or SSA). Three angles alone are insufficient.
What is the SSA ambiguous case?
With two sides and a non-included angle, there may be two, one, or no triangles. The solver detects and shows all valid outcomes.
Can I work in radians?
Yes—toggle Degrees/Radians. Inputs and outputs update immediately.
Is this private?
Yes. Everything runs locally; nothing is uploaded.