Law of Sines Calculator: Solve AAS, ASA and SSA (Ambiguous Case) Triangles
Choose what you know (two angles and a side, or two sides and an angle not between them). You get every side and angle, the area and a scaled drawing.
- Nothing you type is stored
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Sides
a = 10, b = 13.473, c = 15.3209
- Angles
- A = 40°, B = 60°, C = 80°
- Area
- 66.3414
- 2R = a / sin A
- 15.5572
- Perimeter
- 38.7939
a/sin A = b/sin B = c/sin C = 15.5572, the diameter of the circumcircle.
Show working
- C = 180° − 40° − 60° = 80°
- k = a / sin A = 10 / sin 40° = 15.557238
- a = k·sin A = 10, b = k·sin B = 13.472964, c = k·sin C = 15.320889
- Area = ½·b·c·sin A = 66.341395
How to use
- Pick AAS, ASA or SSA depending on what you know.
- Enter the angles in degrees and the side lengths in any one unit.
- Read the sides, angles and area; check “Show working”.
Good to know
Sine rule: a/sin A = b/sin B = c/sin C = 2R, where R is the radius of the circle through the three corners. Once one side–angle pair is known, every other side follows.
SSA is the “ambiguous case”: given a, b and angle A, sin B = b·sin A / a may give two possible angles B (acute and obtuse), one, or none. The calculator lists both triangles when two exist.
Labels: side a is opposite angle A, b opposite B, c opposite C. Angles are in degrees; sides can be in cm, m or any unit, and the area is in that unit squared.
Questions
When do I use the law of sines?
When you know two angles and one side (AAS or ASA), or two sides and an angle opposite one of them (SSA).
What is the ambiguous case?
With SSA, the given side can swing to two positions, making two different triangles. It happens when b·sin A < a < b with A acute.
Why does it say no triangle exists?
The side opposite the given angle is too short to reach the third side (b·sin A > a), or the angles add to 180° or more.
What is 2R in the result?
a/sin A, which equals the diameter of the triangle’s circumscribed circle.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.