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Law of Sines Calculator

Solve triangles fast using the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Includes a SSA ambiguous case checker (0 / 1 / 2 solutions), a clean triangle diagram, and student-friendly step-by-step.

Opposite pairs (super important)

Angle A is opposite side a, angle B opposite b, angle C opposite c. If you mismatch opposite pairs, you’ll get nonsense.

Background

The Law of Sines connects each side of a triangle to the sine of its opposite angle: a/sin(A)=b/sin(B)=c/sin(C). It’s the go-to tool when you know ASA/AAS (two angles and a side) or SSA (two sides and a non-included angle), where the ambiguous case can produce 0, 1, or 2 valid triangles.

Enter values

Tip: Most triangles are in degrees. Radians are common in calculus/physics.

Rounding affects display only.

ASA / AAS inputs

We’ll compute C = 180° − A − B (or π − A − B in radians), then use the Law of Sines to find the missing sides.

SSA inputs (ambiguous case)

This is the given angle (not included between the two sides).

SSA “height” check

Compute h = b·sin(A). Then compare a to h and b:
• if a < h → 0 triangles
• if a = h → 1 right triangle
• if h < a < b → 2 triangles
• if a ≥ b → 1 triangle

We’ll compute sin(B) = b·sin(A)/a. If valid, you may get two angles: B and 180°−B (or π−B).

Given A, a, B → find b

Uses b = a·sin(B)/sin(A). Also computes C and the remaining sides if possible.

Given A, a, b → find B

Important

sin⁻¹ returns the principal angle. In SSA cases, a second angle may exist: B₂ = 180° − B₁ (or π − B₁).

Uses sin(B) = b·sin(A)/a. If valid, you may get two solutions.

Options

Chips prefill and calculate immediately.

Result

No results yet. Enter values and click Calculate.

How to use this calculator

  • Pick a mode: Solve triangle, Missing side, or Missing angle.
  • Choose degrees or radians, then enter your given values.
  • For SSA, check the ambiguous-case output to see whether there are 0, 1, or 2 triangles.
  • Use the diagram to visually confirm your opposite pairs: A ↔ a, B ↔ b, C ↔ c.

How this calculator works

  • Law of Sines: a/sin(A)=b/sin(B)=c/sin(C).
  • Angle sum: A+B+C=180° (or π in radians).
  • SSA ambiguity: because sin(θ)=sin(180°−θ), SSA can yield two triangles.

Formula & Equation Used

Law of Sines: a/sin(A)=b/sin(B)=c/sin(C)

Angle sum: C = 180° − A − B (or C=π−A−B)

SSA helper: sin(B)=b·sin(A)/a, with possible B₂=180°−B₁

Example Problem & Step-by-Step Solution

Example 1 — ASA

Given A=45°, B=65°, and a=12. Solve the triangle.

  1. Compute the third angle: C=180°−45°−65°=70°.
  2. Compute the common ratio: k=a/sin(A)=12/sin(45°).
  3. Find missing sides: b=k·sin(B), c=k·sin(C).

Example 2 — SSA (two solutions possible)

Given A=30°, a=10, b=14. Determine how many triangles exist, then solve.

  1. Compute h=b·sin(A)=14·sin(30°)=7.
  2. Since h<a<b (7 < 10 < 14), there are two possible triangles.
  3. Compute sin(B)=b·sin(A)/a, then B₁=sin⁻¹(...) and B₂=180°−B₁.
  4. For each solution, compute C=180°−A−B, then c=a·sin(C)/sin(A).

Example 3 — Missing side (A, a, B → b)

Given A=40°, a=9, and B=65°. Find side b.

  1. Start with the Law of Sines proportion: a/sin(A)=b/sin(B).
  2. Solve for b: b = a·sin(B)/sin(A).
  3. Plug in: b = 9·sin(65°)/sin(40°).
  4. Compute: b ≈ 12.68.

Quick sanity check: since B is bigger than A, side b should be bigger than a — and it is.

Common mistakes

  • Opposite-pair mismatch: using a with B, etc.
  • SSA trap: forgetting the possible second angle 180°−θ.
  • Radians vs degrees: typing degrees while your calculator is in radians (or vice versa).

Frequently Asked Questions

Q: When is the Law of Sines the best tool?

When you have ASA/AAS or SSA. If you have SAS or SSS, Law of Cosines is often the better starting point.

Q: Why does SSA sometimes give two triangles?

Because sin(θ)=sin(180°−θ). So an arcsin step can produce a second valid angle.

Q: How do I classify the triangle?

If one angle is 90° → right. If all angles < 90° → acute. If one angle > 90° → obtuse.