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
This trigonometric triangle solver helps you quickly solve a triangle when you know some sides and angles but not all. Instead of doing the full algebra by hand, you can plug in the values you have and get the missing sides, missing angles, and area in seconds. It is a practical, student-friendly way to work through triangle geometry, trig homework, construction measurements, and any situation where you need accurate triangle dimensions.
The calculator is built around the two core rules of trigonometry: the Law of Sines and the Law of Cosines. These formulas connect side lengths and angles, letting you “solve” the triangle once enough information is known. The tool also uses the triangle angle sum (A + B + C = 180°) to fill in the final angle, and it computes area using Heron’s formula or the sine area formula when appropriate.
How to use the solver:
- Enter any three values with at least one side (SSS, SAS, ASA, AAS, or SSA).
- Make sure your angle unit matches your inputs (degrees or radians).
- Click Calculate to see all missing sides and angles, plus a step-by-step breakdown.
- Review the results and the triangle diagram to confirm everything looks reasonable.
If you enter an SSA case (two sides and a non-included angle), the tool automatically checks for the ambiguous case and shows zero, one, or two valid triangles. This is a common point of confusion in trigonometry classes, and seeing both solutions helps build intuition about when multiple triangles are possible.
Common real-world uses include estimating roof pitches, checking the dimensions of a triangular truss, verifying a survey measurement, or solving geometry problems in physics and engineering. It can also serve as a triangle side calculator or triangle angle calculator when you are double-checking work by hand. For learners, the visible formulas and steps make it easier to understand why the answer is correct, not just what the answer is.
Quick reference formulas:
- Law of Cosines:
a² = b² + c² − 2bc·cos(A) - Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C) - Area:
Area = ½ab·sin(C)orArea = √(s(s−a)(s−b)(s−c))withs=(a+b+c)/2
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.