Math
Law of Sines Calculator
The law of sines links each side of a triangle to the sine of its opposite angle.
Use it to find a missing side when you know a side and two angles, or a missing angle when you know two sides and one opposite angle. The angle mode checks the ambiguous case and returns both triangles when two exist.
A side whose opposite angle you know.
Try an example
Result
Answer
—
- Common ratio a / sin A
- —
- Third angle and side
- —
- Second solution (if ambiguous)
- —
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Study path
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What the law of sines says
In any triangle the ratio of a side to the sine of its opposite angle is the same for all three pairs. That ratio also equals the diameter of the circumscribed circle. The law of sines is the tool for ASA, AAS, and SSA triangles; SAS and SSS need the law of cosines.
The ambiguous case (SSA)
Solving for an angle can give two answers because sin B = sin(180° − B). Whether the second angle is real depends on how side a compares with the altitude h = b·sin A and with side b:
| Condition (A acute) | Number of triangles |
|---|---|
| a < b·sin A | 0 — side a cannot reach |
| a = b·sin A | 1 — right triangle at B |
| b·sin A < a < b | 2 — B acute or obtuse |
| a ≥ b | 1 — only the acute B fits |
| A ≥ 90° and a ≤ b | 0 |
| A ≥ 90° and a > b | 1 |
How to use it
- Choose whether you are solving for a side or an angle.
- Enter side a and its opposite angle A — this pair sets the common ratio.
- Enter the angle opposite the side you want, or the side opposite the angle you want.
- Read the answer, the third angle and side, and the second solution if the case is ambiguous.
How to read the answer
The common ratio a / sin A is the same for every side-angle pair in the triangle, so once you have it, any side is ratio × sin(opposite angle). When solving for an angle, an inverse sine always returns the acute option; the calculator tells you whether the obtuse option 180° − B also produces a valid triangle.
Common mistakes and edge cases
- Pairing a side with the wrong angle. Each side goes with the angle opposite it, not the angle next to it.
- Forgetting the second angle in the SSA case. sin⁻¹ only returns angles up to 90°.
- Two known angles adding to 180° or more — no third angle is left.
- Using the law of sines for SAS or SSS. Those need the law of cosines first.
- Working in An angle unit where a full turn is 2π instead of 360 degrees. on a calculator set to degrees, or the reverse. This page uses degrees.
Worked examples
Find b from a = 8, A = 40°, B = 65°
Answer
b = 11.2797
Find B from a = 8, A = 40°, b = 10
Ambiguous — two triangles
Answer
B = 53.4641°
Find B from a = 12, A = 40°, b = 10
a ≥ b, one triangle
Answer
B = 32.3884°
Right triangle case: a = 5, A = 30°, b = 10
sin B = 1
Answer
B = 90°
No solution: a = 3, A = 40°, b = 10
sin B > 1
Answer
—
Angles too large: A = 100°, B = 90°
Answer
—
Decimal: a = 6.5, A = 48.2°, B = 71.9°
Answer
b = 8.2878
Frequently asked questions
When do I use the law of sines instead of the law of cosines?+
Use the law of sines when you know a side and its opposite angle (ASA, AAS, SSA). Use the law of cosines for SAS and SSS, where no side-opposite-angle pair is known yet.
Why does the law of sines give two answers?+
Because sin B = sin(180° − B). When solving for an angle from SSA data, both the acute and obtuse angles can be valid if each leaves room for the third angle.
How do I know if the ambiguous case applies?+
Compare side a (opposite the known angle) with b·sin A and with b. Two triangles exist only when b·sin A < a < b and A is acute.
Can I use the law of sines for a right triangle?+
Yes, and it reduces to SOH-CAH-TOA because sin 90° = 1: a / sin A = c.
What does the common ratio equal?+
a / sin A equals 2R, twice the circumradius — the diameter of the circle through the triangle's three vertices.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026