Law of Sines Calculator

Solve triangles using the Law of Sines. Handles SSA and SAA configurations, detects the ambiguous case, and shows all solutions with step-by-step working.

📐 Law of Sines Calculator

Enter side a (opposite angle A), side b, and angle A

°

Enter side a, angle A (opposite a), and angle B

°
°
Angle A
—
Angle B
—
Angle C
—
Side a
—
Side b
—
Side c
—
Triangle Area
—
Sine Ratio (a/sin A)
—

Step-by-step working

📐 What is the Law of Sines?

The Law of Sines (also called the Sine Rule) is a fundamental trigonometric relationship that applies to every triangle, not just right triangles. It states that the ratio of a side length to the sine of its opposite angle is constant throughout the triangle: a/sin(A) = b/sin(B) = c/sin(C). This common ratio equals 2R, where R is the circumradius - the radius of the circle that passes through all three vertices of the triangle.

The Law of Sines is the primary tool for solving triangles in two important situations. The first is SAA (or AAS): when you know one side and two angles, the third angle is found by subtracting from 180°, and the Law of Sines gives the remaining sides. This always has a unique solution. The second is SSA: when you know two sides and a non-included angle - the famous ambiguous case, which may produce zero, one, or two valid triangles depending on the specific values.

Real-world applications are extensive. In surveying and navigation, the Law of Sines is used for triangulation - computing distances to an inaccessible point by observing angles from two known locations. In architecture and structural engineering, it helps calculate forces and lengths in truss structures. In astronomy, it underpins stellar parallax calculations. In physics, oblique collisions and wave refraction problems use the same sine ratio relationships.

This calculator handles both SAA and SSA configurations. For SSA (the ambiguous case), it detects and displays both solutions when they exist - a feature many basic calculators miss. The step-by-step working shows the sine ratio calculation and angle sum verification so you can follow the method and use it in your own work.

📐 Formula

a / sin(A) = b / sin(B) = c / sin(C) = 2R
a, b, c = side lengths of the triangle
A, B, C = angles opposite to sides a, b, c respectively
R = circumradius (radius of the circumscribed circle)
Angle sum: A + B + C = 180°
Area: = ½ × a × b × sin(C)
Example: If a = 7, A = 40°, B = 60°: b = 7 × sin(60°) / sin(40°) ≈ 9.42, C = 80°, c = 7 × sin(80°) / sin(40°) ≈ 10.73

📖 How to Use This Calculator

Steps

1
Choose a configuration - Select SSA if you know two sides and a non-included angle (side a, side b, and angle A). Select SAA if you know one side and two angles (side a, angle A, and angle B).
2
Enter the known values - Type the values in the input fields. All angles must be in degrees and all sides must be positive. For SAA, angles A and B must sum to less than 180°.
3
Read all triangle parts - The calculator shows all three angles, all three sides, the triangle area, and the common sine ratio a/sin(A). If the ambiguous SSA case applies, a second solution is shown below the main result.

💡 Example Calculations

Example 1 — SAA: One Side and Two Angles

Triangle with a = 8, A = 45°, B = 60°

1
Find C: C = 180° − 45° − 60° = 75°
2
Find b: b = a × sin(B) / sin(A) = 8 × sin(60°) / sin(45°) = 8 × 0.8660 / 0.7071 = 9.7980
3
Find c: c = 8 × sin(75°) / sin(45°) = 8 × 0.9659 / 0.7071 = 10.9282
Angles: 45°, 60°, 75°  |  Sides: 8, 9.7980, 10.9282  |  Area = ½ × 8 × 9.7980 × sin(75°) ≈ 37.8564
Try this example →

Example 2 — SSA: Ambiguous Case (Two Solutions)

Triangle with a = 7, b = 10, A = 40°

1
sin(B) = b × sin(A) / a = 10 × sin(40°) / 7 = 10 × 0.6428 / 7 = 0.9182
2
Since 0 < sin(B) < 1, two angles satisfy this: B₁ = arcsin(0.9182) ≈ 66.7°, and B₂ = 180° − 66.7° = 113.3°
3
Solution 1: C₁ = 180°−40°−66.7° = 73.3°, c₁ ≈ 9.66. Solution 2: C₂ = 180°−40°−113.3° = 26.7°, c₂ ≈ 4.55
Two valid triangles exist. The ambiguous case applies - check both solutions.
Try this example →

Example 3 — SSA: Unique Solution (a ≥ b)

Triangle with a = 12, b = 8, A = 50°

1
Since a ≥ b (12 ≥ 8), there is at most one solution. sin(B) = 8 × sin(50°) / 12 = 8 × 0.7660 / 12 = 0.5107
2
B = arcsin(0.5107) ≈ 30.7°. C = 180° − 50° − 30.7° = 99.3°
3
c = 12 × sin(99.3°) / sin(50°) = 12 × 0.9875 / 0.7660 = 15.47
Unique solution: B ≈ 30.7°, C ≈ 99.3°, c ≈ 15.47  |  Area ≈ 36.67
Try this example →

❓ Frequently Asked Questions

What is the Law of Sines?+
The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C) = 2R for any triangle, where a, b, c are sides, A, B, C are opposite angles, and R is the circumradius. It holds for all triangles — acute, obtuse, and right. The law derives from the fact that the same arc subtends equal inscribed angles in a circle.
When should I use the Law of Sines vs the Law of Cosines?+
Use the Law of Sines for SAA/AAS (one side and two angles) and SSA (two sides and a non-included angle). Use the Law of Cosines for SAS (two sides and the included angle, a² = b² + c² − 2bc cos A) and SSS (all three sides). The mnemonic: if you have an angle–side pair plus one more piece, use the Law of Sines.
What is the ambiguous case of the Law of Sines?+
The ambiguous case occurs with SSA: given sides a, b and angle A (opposite to a). If a < b sin(A): no triangle. If a = b sin(A): one right triangle. If b sin(A) < a < b: two triangles (both B ≈ arcsin(sin B) and 180° − B are valid). If a ≥ b: one triangle. The “ambiguous” label comes from the fact that sin(B) = sin(180°−B), giving two possible angles.
How do you find the missing side using the Law of Sines?+
Set up the proportion: a/sin(A) = x/sin(X), where x is the unknown side and X is its opposite angle. Solve: x = a × sin(X) / sin(A). Example: a = 10, A = 30°, B = 70° → b = 10 × sin(70°) / sin(30°) = 10 × 0.9397 / 0.5 = 18.79. Always identify which side and angle are opposite each other.
What does 2R mean in the Law of Sines?+
The common ratio a/sin(A) = b/sin(B) = c/sin(C) equals 2R, where R is the circumradius — the radius of the circle passing through all three triangle vertices (the circumscribed circle). This is the Inscribed Angle Theorem: an inscribed angle is half the central angle subtending the same arc. So a = 2R sin(A), directly relating side length to the circumradius and opposite angle.
How is the Law of Sines used in navigation and surveying?+
Triangulation uses the Law of Sines to find the distance to an inaccessible point P from two known positions A and B. Measure the baseline AB and the angles PAB and PBA. Then PA/sin(PBA) = AB/sin(APB). This technique was used to survey entire countries in the 18th and 19th centuries, and is the basis of GPS triangulation. Sailors used it to find their distance from a lighthouse by measuring horizontal angles.
Can the Law of Sines be used for obtuse triangles?+
Yes — the Law of Sines works for all triangles including obtuse ones (with one angle > 90°). Note that sin(A) = sin(180°−A), so an obtuse angle and its supplement have the same sine. This is exactly why the SSA ambiguous case can produce two solutions: the obtuse and acute values of B give the same sin(B). The formula a/sin(A) = b/sin(B) holds regardless of whether A, B, C are acute or obtuse.
How do you compute triangle area with the Law of Sines?+
Area = ½ × a × b × sin(C) for any two sides and their included angle. Using the Law of Sines, you can also write Area = a² sin(B) sin(C) / (2 sin(A)), expressing area entirely in terms of one side and two angles. This calculator computes area from the two known sides (or derived sides) and the included angle once the full triangle is solved.
What is the SSS (three sides) case and why does the Law of Sines not apply directly?+
With SSS, you know all three sides but no angle. The Law of Sines needs at least one angle (because it is a ratio involving sin(angle)). Use the Law of Cosines instead: cos(A) = (b² + c² − a²) / (2bc) to find one angle, then use the Law of Sines for the remaining angles. Alternatively, use the Law of Cosines for all three angles independently.
How do you verify a Law of Sines solution?+
Check three things: (1) A + B + C = 180° exactly. (2) All sides and angles are positive. (3) The sine ratio is consistent: a/sin(A) = b/sin(B) = c/sin(C). This calculator shows the common ratio a/sin(A) for direct verification. A quick area cross-check also works: ½ab sin(C) should equal ½ac sin(B) and ½bc sin(A).
Tags: Law of Sines Calculator Sine Rule Calculator SSA Triangle Solver SAA Triangle Ambiguous Case Triangle Solver Trigonometry Calculator A/SinA Law of Sines Formula Oblique Triangle

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What is the Law of Sines?

The Law of Sines states that in any triangle, a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are side lengths and A, B, C are the opposite angles. This common ratio equals 2R, where R is the circumradius (radius of the circumscribed circle). The law holds for all triangles - acute, obtuse, and right.

When do you use the Law of Sines?

Use the Law of Sines when you know: (1) SAA/AAS - one side and two angles (always has a unique solution), or (2) SSA - two sides and a non-included angle (the ambiguous case - may have 0, 1, or 2 solutions). You cannot use it for SAS (two sides and the included angle) or SSS (all three sides) - use the Law of Cosines instead.

What is the ambiguous case (SSA)?

The ambiguous case occurs in the SSA configuration where you know sides a, b and angle A (A is opposite to a). Depending on the values, there may be: no triangle (if a < b sin A), one right triangle (if a = b sin A), one triangle (if a ≥ b), or two triangles (if b sin A < a < b). This calculator detects and shows both solutions when the two-triangle case applies.

What is the difference between the Law of Sines and Law of Cosines?

The Law of Sines (a/sin A = b/sin B = c/sin C) works when you have an angle-side pair plus one more piece of information. The Law of Cosines (a² = b² + c² − 2bc cos A) works when you have three sides (SSS) or two sides and the included angle (SAS). For SAA/AAS or SSA, use the Law of Sines first; for SAS/SSS, use the Law of Cosines.

How do you find the area using the Law of Sines?

The area of a triangle can be found as: Area = ½ × a × b × sin(C), where a and b are two sides and C is the included angle between them. This follows directly from the standard area formula (base × height / 2) combined with trigonometry. The formula works for any pair of sides as long as you know the included angle.

What is the circumradius formula from the Law of Sines?

The Law of Sines gives 2R = a/sin(A), where R is the circumradius (radius of the circumscribed circle passing through all three vertices). So R = a / (2 sin A). This means the sine ratio a/sin(A) shown by this calculator equals 2R - a direct geometric interpretation of the law.

Can the Law of Sines solve right triangles?

Yes - for a right triangle with the right angle at C (C = 90°), sin(C) = 1, so the formula becomes a/sin(A) = b/sin(B) = c. This simplifies to sin(A) = a/c and sin(B) = b/c, which are just the standard SOHCAHTOA definitions. However, for right triangles, basic trig is simpler. The Law of Sines is most valuable for oblique (non-right) triangles.

What does SSA, SAA, AAS, and SAS mean in triangle solving?

These abbreviations describe which three of the six triangle parts (three sides and three angles) are known. S = side, A = angle. SSA: two sides and a non-included angle (ambiguous case). SAA or AAS: one side and two angles. SAS: two sides and the included angle (use Law of Cosines). SSS: all three sides (use Law of Cosines). The Law of Sines applies to SSA and SAA/AAS.

How do you verify a Law of Sines solution?

Verify by checking: (1) all three angles sum to 180°, (2) all sides and angles are positive, (3) the sine ratios a/sin(A) = b/sin(B) = c/sin(C) are all equal, and (4) the area computed from different pairs (½ab sin C = ½ac sin B = ½bc sin A) gives the same result. This calculator shows the common sine ratio for easy verification.

📌 Quick Tips

💡The Law of Sines applies to ALL triangles (not just right triangles) and is the primary tool when you know two angles and any side (SAA/AAS), or two sides and a non-included angle (SSA).
💡The ambiguous case (SSA) can produce 0, 1, or 2 valid triangles. When given a, b, and angle A: if a ≥ b, there is exactly one triangle; if a < b sin(A), there is no triangle; if b sin(A) < a < b, there are two triangles.
💡Always verify your answer: the sine ratio a/sin(A) must equal b/sin(B) must equal c/sin(C) for a valid solution. This calculator shows the ratio for cross-checking.
💡Use the Law of Cosines instead when given SAS (two sides and the included angle) or SSS (all three sides). The Law of Sines requires at least one known angle.
💡The FAQ section below answers the most common questions about this topic.