What is the Law of Sines?
The Law of Sines relates each side of a triangle to the sine of its opposite angle. For any ordinary triangle:
a / sin(A) = b / sin(B) = c / sin(C)The lowercase side labels are paired with the uppercase angles directly opposite them: side a is opposite angle A, side b is opposite B, and side c is opposite C. Keeping those opposite pairs matched is the most important setup step.
When should you use the Law of Sines?
The Law of Sines is especially useful when you know at least one complete side-angle pair. Common solvable configurations are:
- ASA: two angles and the included side.
- AAS: two angles and a non-included side.
- SSA: two sides and a non-included angle, which can produce zero, one or two triangles.
If you know SAS or SSS and do not yet have an opposite side-angle pair, the Law of Cosines is usually the natural starting point instead.
How to solve an ASA or AAS triangle
When two angles are known, the third angle comes from the triangle angle sum:
C = 180° − A − BOnce all three angles are known and one side is known, use the common sine ratio to obtain the missing sides.
Example: A = 45°, B = 65°, and a = 12. First:
C = 180° − 45° − 65° = 70°Then:
b = 12 × sin(65°) / sin(45°) c = 12 × sin(70°) / sin(45°)This produces a unique triangle because two angles already fix the shape.
What is the SSA ambiguous case?
SSA means two sides and an angle opposite one of those sides are known. Unlike ASA or AAS, this information can describe no triangle, exactly one triangle, or two different triangles.
The ambiguity occurs because sine has the same positive value for supplementary angles:
sin(θ) = sin(180° − θ)So when the Law of Sines gives sin(B), the angle B may be either the principal arcsine value or its supplement—provided the remaining angle is still positive.
SSA example with two valid triangles
Suppose A = 30°, a = 10 and b = 14. The Law of Sines gives:
sin(B) = b sin(A) / a = 14 × 0.5 / 10 = 0.7The first possible angle is:
B₁ = sin⁻¹(0.7) ≈ 44.427°The second possible angle is:
B₂ = 180° − 44.427° ≈ 135.573°Because A + B₂ is still less than 180°, both create valid triangles. The calculator returns both complete solutions rather than silently choosing one.
SSA height test
For the common setup where angle A is acute and side a is opposite A while b is the other known side, a useful geometric height is:
h = b sin(A)When a < b:
- if a < h, there is no triangle;
- if a = h, there is one right triangle;
- if h < a < b, there are two triangles.
If a ≥ b, the acute-angle SSA setup gives one triangle. The calculator does not rely only on this shortcut; it validates candidate angles directly, which also handles obtuse given angles correctly.
Why obtuse SSA needs care
If the given angle A is obtuse, the side opposite it must be the longest side. Therefore, if A is greater than 90° and a is not longer than b, no triangle can exist.
This is another reason a robust SSA solver should validate the full angle and side geometry instead of applying an acute-angle shortcut blindly.
How to find a missing side
If A, a and B are known, solve for b:
b = a sin(B) / sin(A)After finding C = 180° − A − B, side c is:
c = a sin(C) / sin(A)The same pattern works with any correctly matched known side-angle pair.
How to find a missing angle
If A, a and b are known:
sin(B) = b sin(A) / aIf that ratio is greater than 1, there is no real angle B and therefore no triangle. If it lies between 0 and 1, take arcsine and then test the supplementary angle as a second possibility.
Why inverse sine alone is not enough for SSA
A calculator that returns only B = arcsin(x) can miss the second valid triangle. The supplementary candidate 180° − B has the same sine.
The second candidate is valid only if all three triangle angles remain positive. SonoCalculator explicitly checks this condition and reports the number of valid solutions.
Area of a solved triangle
Once two sides and their included angle are known:
Area = ½ab sin(C)Equivalent forms are ½bc sin(A) and ½ca sin(B). The calculator uses the solved triangle values, so area is available for ASA, AAS and every valid SSA solution.
Perimeter
The perimeter is simply:
P = a + b + cIn an SSA ambiguous case, the two valid triangles generally have different missing sides and therefore different perimeters and areas.
Circumradius and the extended Law of Sines
The Law of Sines also connects the common side-to-sine ratio to the circumradius R:
a / sin(A) = b / sin(B) = c / sin(C) = 2RTherefore:
R = a / (2 sin A)The circumradius is the radius of the circle passing through all three triangle vertices.
Triangle classification
A solved triangle can be classified by its angles and side lengths. By angles, a triangle is acute, right or obtuse. By sides, it is scalene, isosceles or equilateral.
Because decimal input and trigonometric computation involve rounding, the calculator uses a small tolerance when deciding whether angles or sides should be treated as equal.
ASA versus AAS
ASA and AAS use the same Law of Sines workflow after the third angle is found. The difference is only where the known side sits relative to the two known angles.
Both configurations determine a unique triangle as long as the two entered angles are positive and add to less than 180°.
SSA versus SAS
SSA gives an angle opposite one of the two known sides and may be ambiguous. SAS gives the angle included between the two known sides and produces a unique triangle when valid.
SAS typically begins with the Law of Cosines because there is not yet a complete opposite side-angle pair. Confusing SSA and SAS is a common source of incorrect solutions.
Why side-angle pairing matters
The Law of Sines does not match any side with any angle. Each side must be paired with its opposite angle:
a ↔ A, b ↔ B, c ↔ CIf side a is accidentally paired with sin(B), the resulting ratio no longer represents the same triangle geometry.
Can the Law of Sines solve a right triangle?
Yes. The Law of Sines is valid for right triangles too. If C = 90°, then sin(C) = 1, and:
c / sin(90°) = cIf c is the hypotenuse, this is consistent with familiar right-triangle sine relationships.
Can the Law of Sines solve SSS?
Not as the natural first step. With three sides and no angles, there is no known side-angle pair. The Law of Cosines is usually used first to obtain an angle, after which the Law of Sines can help find another angle if desired.
Can the Law of Sines solve SAS?
Again, not usually as the first step. SAS supplies two sides and the included angle, but that angle is not opposite either known side. The Law of Cosines gives the third side directly, creating the side-angle pair needed for later Law of Sines work.
Degrees and radians
This calculator uses degrees because they are the most common format in school triangle problems and make the 180° angle sum explicit. The mathematical law itself is independent of angle unit, provided the trigonometric functions and angle sums use the same convention.
Precision and rounding
Premature rounding can noticeably change the final missing side or angle, especially in long calculations or near an ambiguous-case boundary. The calculator keeps internal floating-point precision and rounds only the displayed values.
For hand calculations, retain several extra digits in intermediate sine and inverse-sine values before rounding the final answer.
Near-degenerate triangles
If two angles nearly add to 180°, the third angle becomes very small. The resulting triangle can be extremely thin, and small input changes may cause large relative changes in some outputs.
A mathematically positive third angle can still describe a numerically sensitive triangle. Use appropriate measurement precision for real-world work.
Law of Sines in surveying and navigation
Triangle solving appears in surveying, navigation and triangulation when angles and a baseline or other side are known. The Law of Sines can convert angular observations into distances in an idealized planar triangle.
Large-scale geodesy, Earth curvature, map projections and measurement uncertainty require methods beyond a simple Euclidean triangle solver.
Law of Sines in physics and engineering
Oblique triangles appear in force diagrams, geometry and vector-related problems. The Law of Sines can solve a triangle once the given quantities truly form side-angle geometry.
Units for all sides must be consistent. Angles are dimensionless geometrically but must be entered in the calculator's expected degree format.
Common mistakes with the Law of Sines
- Pairing the wrong side and angle. Always match opposite elements.
- Forgetting that triangle angles sum to 180°.
- Missing the second SSA solution. Test the supplementary angle.
- Accepting arcsin of a ratio greater than 1. That means the requested triangle is impossible.
- Using Law of Sines first for SSS or SAS. Law of Cosines is generally more direct there.
- Rounding too early. Preserve intermediate precision.
- Treating a sketch as proof of scale. A diagram is only a visual aid unless explicitly constructed to scale.
- Ignoring near-degenerate geometry. Very small angles can amplify measurement error.
How to verify a solved triangle
First confirm:
A + B + C = 180°Then compare all three side-to-sine ratios:
a/sin(A), b/sin(B), c/sin(C)They should agree within rounding tolerance. You can also verify the area using more than one equivalent formula, such as ½ab sin(C) and ½bc sin(A).
Frequently asked questions
What is the Law of Sines formula?
a/sin(A) = b/sin(B) = c/sin(C).
When should I use the Law of Sines?
Use it when you have a known opposite side-angle pair, especially in ASA, AAS and SSA problems.
What is the ambiguous case?
SSA can sometimes produce zero, one or two valid triangles because an angle and its supplement have the same sine.
Can SSA have two answers?
Yes. If both the principal inverse-sine angle and its supplement leave a positive third angle, two triangles exist.
How do I know there is no SSA triangle?
If the required sine value exceeds 1 in magnitude, or the candidate angles cannot sum to less than 180°, no triangle exists.
Does ASA have an ambiguous case?
No. Two valid angles determine the third angle and therefore the triangle shape uniquely.
What is the extended Law of Sines?
It adds the circumradius relationship: a/sin(A) = b/sin(B) = c/sin(C) = 2R.
Can I use this on a right triangle?
Yes. The Law of Sines remains valid for right triangles.
Why does the calculator show two solution tabs?
Your SSA inputs admit two distinct valid triangles, so both must be shown rather than one being discarded.
Why is there no result for my values?
The inputs may be invalid—for example, nonpositive sides, angles outside 0°–180°, an angle sum of 180° or more, or SSA geometry with no possible triangle.
Final note: the Law of Sines is most reliable when opposite pairs are labeled before any arithmetic begins. For ASA and AAS, solve the third angle and use the common sine ratio. For SSA, never stop at the first inverse-sine result—test the supplementary angle and validate the remaining angle. This calculator keeps every valid solution visible so the ambiguous case cannot be missed.