Law of Sines Calculator

About Law of Sines Calculator

Solve any triangle using the law of sines: a/sin(A) = b/sin(B) = c/sin(C). Enter known sides and angles to find all missing values. Handles the ambiguous SSA case by checking for zero, one, or two valid solutions.

The Law of Sines Formula

In any triangle, the ratio of each side to the sine of its opposite angle is constant:

a/sin(A) = b/sin(B) = c/sin(C) = 2R

where R is the circumradius (radius of the circumscribed circle). This relationship lets you find unknown sides and angles when you have enough information.

When to Use the Law of Sines

Known Information	Case Name	Use Law of Sines?
Two angles + any side	AAS or ASA	Yes - straightforward, one solution
Two sides + non-included angle	SSA	Yes - but check for ambiguous case
Two sides + included angle	SAS	No - use law of cosines
Three sides	SSS	No - use law of cosines

Worked Example: AAS Case

Given: A = 40°, B = 60°, a = 10

1. Find angle C: C = 180° - 40° - 60° = 80°
2. Set up the ratio: 10/sin(40°) = b/sin(60°) = c/sin(80°)
3. Calculate the common ratio: 10/sin(40°) = 10/0.6428 = 15.557
4. Find b: b = 15.557 × sin(60°) = 15.557 × 0.8660 = 13.47
5. Find c: c = 15.557 × sin(80°) = 15.557 × 0.9848 = 15.32

Verify: 10/sin(40°) = 13.47/sin(60°) = 15.32/sin(80°) ≈ 15.557 ✓

The Ambiguous Case (SSA)

When you know two sides and a non-included angle, there may be zero, one, or two valid triangles. This is the trickiest scenario in triangle solving.

How to determine the number of solutions:

Given: side a, side b, and angle A (where A is opposite side a):

Condition	Solutions	Explanation
a < b × sin(A)	0	Side a is too short to form any triangle
a = b × sin(A)	1	Side a exactly reaches - creates a right triangle
b × sin(A) < a < b	2	Side a can swing to two valid positions
a ≥ b	1	Only one triangle is possible

Worked example (two solutions): Given a = 8, b = 12, A = 30°

1. sin(B) = b × sin(A) / a = 12 × sin(30°) / 8 = 12 × 0.5 / 8 = 0.75
2. B could be arcsin(0.75) = 48.59° (acute) or 180° - 48.59° = 131.41° (obtuse)
3. Solution 1: B = 48.59°, C = 180° - 30° - 48.59° = 101.41°, c = 8 × sin(101.41°)/sin(30°) = 15.68
4. Solution 2: B = 131.41°, C = 180° - 30° - 131.41° = 18.59°, c = 8 × sin(18.59°)/sin(30°) = 5.10

Law of Sines vs Law of Cosines

	Law of Sines	Law of Cosines
Formula	a/sin(A) = b/sin(B)	c² = a² + b² - 2ab cos(C)
Best for	AAS, ASA, SSA	SAS, SSS
Complexity	Simpler algebra	More computation
Ambiguity risk	Yes (SSA case)	No ambiguity
Special case	-	Reduces to Pythagorean theorem when C = 90°

Finding Triangle Area with the Law of Sines

Once you have two sides and the included angle, the area is:

Area = (1/2) × a × b × sin(C)

This formula works for any triangle and is a direct consequence of the law of sines relationship.

For SAS and SSS cases, the law of cosines calculator handles those directly. For general triangle properties including area by multiple methods, the triangle calculator covers all the basics.

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