Interactive guide
Three sides, three angles, two laws that connect them
Any triangle has six measurements: three sides, three angles. Know enough of them and the Law of Sines or the Law of Cosines hands you the rest. Below, drag any corner of a real triangle and watch both laws hold exactly, live, no matter the shape. Then see the one case where "enough measurements" doesn't guarantee a single answer.
Two laws, two different jobs
The Law of Sines, a/sin A = b/sin B = c/sin C, says every side's length divided by the sine of its opposite angle gives the same number, no matter which of the three pairs you pick. It's the tool for problems built from angles: given two angles and a side (ASA, AAS), or two sides and a non-included angle (SSA).
The Law of Cosines, c² = a² + b² − 2ab·cos C, is the Pythagorean theorem with a correction term for triangles that aren't right triangles. It's the tool for problems built from sides: given all three sides (SSS), or two sides and the angle between them (SAS), neither of which the Law of Sines can even get started on, since it needs an angle paired with its own opposite side.
Which law, which case
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1
SSS or SAS → start with Cosines
Three sides, or two sides and the angle between them: only the Law of Cosines can use what you have.
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2
ASA or AAS → Sines solves it outright
Two angles plus any side: the third angle is 180° minus the other two, then Sines gives both remaining sides.
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3
SSA → Sines, but check your answer
Two sides and a non-included angle is the one case with 0, 1, or 2 valid triangles. See the second panel below.
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4
Once you have 3 sides + 1 angle, Sines finishes the rest
After Cosines gets you one unknown, switch to Sines for the remaining angles, it's simpler algebra.
Drag any corner. Every side and angle recomputes live.
- a, b, c
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- A, B, C
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Law of Sines: three equal ratios
All three always match, whatever shape the triangle: this common value is its circumdiameter (2R).
Law of Cosines: check any angle from the 3 sides alone
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The SSA ambiguous case
Fix angle A and side b (adjacent to it), then let side a (opposite A) swing freely. Watch how many triangles actually close.
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- h = b·sin A
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solution 1 solution 2
SSA is the one combination of "two sides and an angle" that isn't automatically safe, and the reason is geometric, not algebraic: fixing an angle and its adjacent side pins down two full rays, but the third side is only told its length, not which direction to point. A circle of that length can cross the second ray twice, once, or not at all. Every other case (SAS, ASA, AAS, SSS) pins the triangle down completely, which is exactly why only SSA gets taught with a warning label.
The jargons
Six measurements per triangle, and which subset you're handed determines which law applies and whether the answer is unique.
a/sin A = b/sin B = c/sin C. Connects a side to its own opposite angle; needs at least one matched pair to start.
c² = a²+b²−2ab·cos C. Works from three sides, or two sides and their included angle, no opposite pair required.
Two sides and a non-included angle. Depending on the numbers, 0, 1, or 2 distinct triangles satisfy the same three measurements.
Included: the angle sits between the two given sides (SAS, safe). Non-included: it doesn't (SSA, ambiguous).
The common value every Law of Sines ratio equals: the diameter of the one circle passing through all three vertices.
The shortest possible length side a could have and still reach the second ray at all; below it, no triangle closes.