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The Law of Sines

a/sin(A) = b/sin(B) = c/sin(C). The ratio of a side to the sine of its opposite angle is constant. Solves AAS, ASA, and SSA triangles — including the ambiguous case.

Concept Fundamentals
a/sin(A) = b/sin(B) = c/sin(C)
Formula
AAS, ASA, SSA
Cases
a/sin(A) = 2R
Circumradius
SSA: 0, 1, or 2 triangles
Ambiguous

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ASA and AAS always give a unique triangle. SSA can give 0, 1, or 2 triangles (ambiguous case). If sin(B) > 1 in SSA, no triangle exists. If b·sin(A) < a < b, two triangles may exist. The constant a/sin(A) equals 2R, where R is the circumradius of the triangle.

Key quantities
a/sin(A) = b/sin(B) = c/sin(C)
Formula
Key relation
AAS, ASA, SSA
Cases
Key relation
a/sin(A) = 2R
Circumradius
Key relation
SSA: 0, 1, or 2 triangles
Ambiguous
Key relation

Ready to run the numbers?

Why: The Law of Sines is essential for navigation, surveying, and triangulation when you know angles and need to find sides — or have SSA and must resolve the ambiguous case.

How: Given two angles, find the third (sum = 180°). Then use the ratio a/sin(A) = b/sin(B) = c/sin(C) to find unknown sides. For SSA, solve sin(B) = b·sin(A)/a and check for 0, 1, or 2 solutions.

ASA and AAS always give a unique triangle. SSA can give 0, 1, or 2 triangles (ambiguous case).If sin(B) > 1 in SSA, no triangle exists. If b·sin(A) < a < b, two triangles may exist.
Sources:Wolfram MathWorld — Law of SinesKhan Academy — Law of Sines

Run the calculator when you are ready.

Start CalculatingSelect case (ASA, AAS, SSA) and enter known sides and angles

Examples — Click to Load

Triangle Case

law-of-sines.sh
CALCULATED
$ solve --case ASA --triangle
Side a
10
Side b
14.14
Side c
19.32
∠A
30°
∠B
45°
∠C
105.00°
Sum
180°
Case
ASA
Share:
Law of Sines Result
a=10, b=14.14, c=19.32
∠A=30° ∠B=45° ∠C=105.00°
Case: ASASum: 180°
numbervibe.com/calculators/mathematics/trigonometry/law-of-sines-calculator

Triangle Dimensions

Side Lengths

Angle Proportions

Triangle Visualization

Calculation Breakdown

FORMULA
Law of Sines
a/sin(A) = b/sin(B) = c/sin(C)
ext{Ratio} ext{of} ext{side} ext{to} ext{sine} ext{of} ext{opposite} ext{angle}
SOLUTION
Third Angle C
105°
180° - 30° - 45°
PRIMARY RESULT
SIDE B
14.142136
a·sin(B)/sin(A)
PRIMARY RESULT
SIDE C
19.318517
a·sin(C)/sin(A)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🧭

Mariners and pilots use the Law of Sines for celestial navigation.

— Navigation textbooks

📐

Surveyors use triangulation with the Law of Sines to measure distances across rivers.

— Surveying standards

Key Takeaways

  • • a/sin(A) = b/sin(B) = c/sin(C) — the ratio of a side to the sine of its opposite angle is constant
  • • Use for AAS, ASA, SSA triangles. For SSS or SAS, use the Law of Cosines
  • • SSA is ambiguous: 0, 1, or 2 triangles may exist depending on side lengths
  • • Essential for navigation, surveying, and triangulation when angles are known
  • • The constant equals 2R where R is the circumradius of the triangle

Did You Know?

🧭Mariners and pilots use the Law of Sines for celestial navigation — measuring angles to stars to determine positionSource: Navigation textbooks
📐Surveyors use triangulation with the Law of Sines to measure distances across rivers and valleys without crossing themSource: Surveying standards
🔺The Law of Sines was known to medieval Islamic mathematicians; Nasir al-Din al-Tusi gave a complete proofSource: History of Mathematics
📡GPS and radar systems use triangular geometry and the Law of Sines for position determinationSource: IEEE Signal Processing
🌊In wave physics, the Law of Sines appears in Snell's law for refraction: n₁sin(θ₁) = n₂sin(θ₂)Source: Physics textbooks
🏗️Structural engineers use the Law of Sines to resolve forces in truss systems and determine load distributionSource: Engineering Toolbox

How It Works

ASA & AAS

Given two angles, find the third (sum = 180°). Then use a/sin(A) = b/sin(B) = c/sin(C) to find unknown sides. Each ratio is the same constant 2R.

SSA — The Ambiguous Case

Given sides a, b and angle A: sin(B) = b·sin(A)/a. If sin(B) > 1, no triangle. If sin(B) < 1 and b < a, two triangles may exist (B and 180°−B).

When to Use Law of Sines vs Cosines

Law of Sines: AAS, ASA, SSA. Law of Cosines: SAS, SSS. Prefer Sines when you have angle-side pairs; Cosines when you have sides and an included angle.

Expert Tips

Label Consistently

Side a is opposite angle A. Mixing these up is the #1 error. Use the Sine Calculator to verify values.

SSA: Check Both Solutions

When b·sin(A) < a < b, two triangles exist. Always verify with the triangle inequality and angle sum.

Units Don't Matter

The Law of Sines uses ratios. Sides can be in meters, feet, or any unit — the ratios stay the same.

Circumradius Formula

a/sin(A) = 2R. So R = a/(2·sin(A)). Useful for inscribed circle problems. See Unit Circle.

Comparison: Law of Sines vs Other Methods

FeatureLaw of SinesLaw of CosinesRight Triangle
Best forAAS, ASA, SSASAS, SSSRight angles
Ambiguous caseYes (SSA)NoNo
Requires anglesAt least 21 or 01 (90°)
Formula complexitySimple ratioQuadraticPythagorean
CircumradiusDirect (2R)Indirectc/2
Surveying useCommonCommonLimited
DerivationArea formulaPythagoreanDefinition
When C=90°Still works→ PythagoreanDirect

Frequently Asked Questions

Can the Law of Sines be used for all triangles?

Yes. It applies to acute, right, and obtuse triangles. You need at least one side and its opposite angle, plus one more side or angle.

What is the ambiguous case?

In SSA, when the given angle is acute and the opposite side is shorter than the adjacent side, two triangles can satisfy the conditions. The calculator finds both.

When should I use Law of Cosines instead?

Use Law of Cosines when you have SAS (two sides and included angle) or SSS (all three sides). Law of Sines cannot directly solve those.

What if sin(B) > 1 in SSA?

No triangle exists. The given measurements are impossible — the side opposite the known angle is too short to reach the other side.

How is the Law of Sines derived?

From the area formula: Area = ½·a·b·sin(C) = ½·b·c·sin(A) = ½·c·a·sin(B). Equating and simplifying gives a/sin(A) = b/sin(B) = c/sin(C).

What units should I use?

Any. The Law of Sines uses ratios, so meters, feet, or dimensionless values all work. Angles must be in degrees (or radians consistently).

Can I use it in 3D?

The planar Law of Sines applies to 2D triangles. For spherical triangles, there is a different Law of Sines used in astronomy and navigation.

What is 2R in the formula?

R is the circumradius — the radius of the circle passing through all three vertices. a/sin(A) = b/sin(B) = c/sin(C) = 2R.

Law of Sines by the Numbers

3
Solve Cases (ASA,AAS,SSA)
180°
Angle Sum
2R
Circumradius Link
2
Max SSA Solutions

Disclaimer: Results are for educational use. Verify triangle validity (angle sum 180°, triangle inequality) for engineering or surveying applications.

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