What is the ambiguous case (SSA)?

When you know two sides and an angle opposite one of them, the height geometry can permit two distinct triangles. The calculator checks both the acute solution for angle B and its supplement, reporting two triangles whenever both close validly.

When does SSA have no solution?

If the side opposite the known angle is shorter than the perpendicular height it must reach, no triangle can form. Numerically this happens when b·sin(A)/a exceeds 1, and the calculator returns a "no solution" note.

Why use the law of sines rather than the law of cosines?

The law of sines is ideal when you already have an angle matched with its opposite side (AAS, ASA, or SSA). The law of cosines is the right tool for SAS and SSS, where no such matched angle-side pair is given directly.

Do AAS and ASA ever produce two triangles?

No. Knowing two angles fixes the third, so the triangle shape is fully determined and a single known side scales it uniquely. Only SSA can be ambiguous.

Law of Sines Calculator

Angle A

Angle B

Side a (opposite A)

OutcomeOne unique triangle.

Side a10.0000

Angle A40.00°

Side b13.4730

Angle B60.00°

Side c15.3209

Angle C80.00°

The Law of Sines Calculator solves a triangle from a side paired with its opposite angle. It covers the AAS and ASA configurations, which always yield a single triangle, and the notorious SSA configuration, where the same measurements can describe two different triangles, one triangle, or none at all. The tool reports every valid solution along with a note explaining the outcome.

Formula

a / sin(A) = b / sin(B) = c / sin(C)

a, b, c: Side lengths of the triangle
A, B, C: Interior angles opposite sides a, b, and c (degrees)

How it works

Choose the input pattern: AAS (two angles and a non-included side), ASA (two angles and the side between them), or SSA (two sides and a non-included angle).
The calculator holds the constant ratio a/sin(A) = b/sin(B) = c/sin(C) and rearranges it to find each unknown side or angle, filling in the last angle from the 180-degree sum.
In SSA mode it tests both the acute angle and its obtuse supplement, returning two triangles when both are geometrically valid, one when only the acute solution survives, and a "no solution" note when the side is too short to close the triangle.

Worked example

AAS: angle A = 40°, angle B = 60°, and side a (opposite A) = 10.

Angle C = 180 − 40 − 60 = 80°.
Common ratio k = a / sin(A) = 10 / sin(40°) = 15.557.
Side b = k·sin(60°) = 13.473 and side c = k·sin(80°) = 15.321.

Angle C = 80°, side b = 13.473, side c = 15.321.

Frequently asked questions

What is the ambiguous case (SSA)?: When you know two sides and an angle opposite one of them, the height geometry can permit two distinct triangles. The calculator checks both the acute solution for angle B and its supplement, reporting two triangles whenever both close validly.
When does SSA have no solution?: If the side opposite the known angle is shorter than the perpendicular height it must reach, no triangle can form. Numerically this happens when b·sin(A)/a exceeds 1, and the calculator returns a "no solution" note.
Why use the law of sines rather than the law of cosines?: The law of sines is ideal when you already have an angle matched with its opposite side (AAS, ASA, or SSA). The law of cosines is the right tool for SAS and SSS, where no such matched angle-side pair is given directly.
Do AAS and ASA ever produce two triangles?: No. Knowing two angles fixes the third, so the triangle shape is fully determined and a single known side scales it uniquely. Only SSA can be ambiguous.

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Law of Cosines Calculator Triangle Calculator Right Triangle Calculator Pythagorean Theorem Calculator