When do you use law of sines vs law of cosines?

Use law of sines for AAS, ASA, or SSA. Use law of cosines for SAS or SSS. Law of sines is simpler when you have at least one side-angle pair.

What is the ambiguous case?

SSA (two sides + non-included angle) may produce 0, 1, or 2 valid triangles because sin(B) = sin(180-B). Both angles must be checked to see if they form valid triangles.

How do you solve a triangle with law of sines?

Find the third angle (180 - A - B), then use a/sin(A) = b/sin(B) to find unknown sides. Cross-multiply and solve for each unknown.

Can you use law of sines for right triangles?

Yes, but basic trig (SOH CAH TOA) is simpler. For right triangles, sin(90) = 1, so the formula simplifies to a/sin(A) = hypotenuse.

What is the circumradius?

The circumradius (R) is the radius of the circle through all three vertices. R = a/(2*sin(A)) using any side-angle pair.

How do you find triangle area with law of sines?

Area = (1/2)*a*b*sin(C) using any two sides and their included angle. Alternatively use Heron's formula with all three sides.

What are AAS, ASA, and SSA?

AAS: two angles + non-included side. ASA: two angles + included side. SSA: two sides + non-included angle (ambiguous). AAS and ASA always give one solution.

Free Law of Sines Calculator

Law of Sines Calculator

Q: What is the law of sines?

The law of sines states a/sin(A) = b/sin(B) = c/sin(C) = 2R. The ratio of any side to the sine of its opposite angle is constant for all three pairs in a triangle.

Solve any triangle using the law of sines. Enter AAS, ASA, or SSA values to find all sides, angles, area, perimeter, circumradius, and more. Handles the ambiguous case with step-by-step solutions.

Quick Examples

Two Angles + One Side - Enter Known Values

Angle A (°)

Angle B (°)

Side a

Step-by-Step Solution

Given: Angle A = 40°, Angle B = 60°, Side a = 10

Step 1: Find Angle C = 180° - 40° - 60° = 80°

Step 2: Use Law of Sines: a/sin(A) = b/sin(B)

b = a × sin(B) / sin(A) = 10 × sin(60°) / sin(40°)

b = 13.472964

Step 3: c = a × sin(C) / sin(A) = 15.320889

Triangle Solution

Scalene Acute

Angles

40°

60°

80°

Sides

13.473

15.3209

Properties

Area66.3414

Perimeter38.7939

Circumradius (R)7.7786

Inradius (r)3.4202

Heights

ha13.2683

hb9.8481

hc8.6603

Law of Sines Formula

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

Where a, b, c are sides opposite to angles A, B, C respectively.

How to Use the Law of Sines Calculator

Select Your Configuration

Choose AAS (two angles + non-included side), ASA (two angles + included side), or SSA (two sides + non-included angle). Each mode shows the appropriate input fields. Use preset examples to see how different triangle types work.

Enter Known Values

Type the values you know - angles in degrees and sides in any consistent unit. The calculator validates inputs in real-time and warns if the values cannot form a valid triangle (e.g., angles summing to more than 180°).

Review the Complete Solution

See all three sides, all three angles, area, perimeter, circumradius, inradius, and heights. For SSA (ambiguous case), both possible triangles are shown if two solutions exist. The step-by-step panel shows the exact calculation process.

Why Use Our Law of Sines Calculator

3 Input Configurations

Solve AAS, ASA, and SSA triangles. Each mode guides you to enter exactly the right values for a complete solution.

Complete Triangle Solution

Get all 3 sides, all 3 angles, area, perimeter, circumradius, inradius, and all 3 altitudes in one calculation.

Step-by-Step Solutions

See every calculation step from the given values to the final answer. Perfect for checking homework or learning the process.

Ambiguous Case Handling

SSA mode correctly detects 0, 1, or 2 solutions and shows all valid triangles. No more guessing about the ambiguous case.

6 Preset Examples

30-60-90, 45-45-90, equilateral, obtuse, ambiguous SSA, and navigation examples. Learn by exploring different triangle types.

High Precision

Results calculated to 6 decimal places with clean display. Angles in degrees, sides in your input units. All computation in-browser.

Complete Guide to the Law of Sines

The Law of Sines Formula and What It Means

The law of sines establishes a fundamental relationship in every triangle: the ratio of any side to the sine of its opposite angle is the same for all three side-angle pairs. Written mathematically: a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius (the radius of the circle passing through all three vertices).

This means that longer sides always face larger angles, and the relationship is precisely proportional through the sine function. In a triangle where angle A is twice angle B, side a is not twice side b - it is sin(2B)/sin(B) = 2cos(B) times side b. This non-linear relationship is what makes the law of sines both powerful and sometimes counterintuitive.

Understanding the Ambiguous Case (SSA)

The ambiguous case is the trickiest application of the law of sines. When you know two sides (a and b) and the angle opposite one of them (A), you are solving sin(B) = b*sin(A)/a. Since sine produces the same value for supplementary angles (sin(30°) = sin(150°) = 0.5), there might be two valid angles for B.

Specifically: if b*sin(A)/a is greater than 1, no triangle exists. If it equals exactly 1, there is one right triangle. If it is less than 1, then B could be either arcsin(result) or 180° - arcsin(result). Both values are valid only if each produces a positive third angle C = 180° - A - B. When both work, you have two different triangles that satisfy the given measurements.

Solving Triangles: AAS vs ASA vs SSA

AAS (Angle-Angle-Side) gives you two angles and a side not between them. This is the simplest case: find the third angle (C = 180° - A - B), then use the law of sines twice to find the other two sides. There is always exactly one solution.

ASA (Angle-Side-Angle) gives you two angles and the included side. Again, find the third angle first, then use the law of sines. There is always exactly one solution. SSA (Side-Side-Angle) is the ambiguous case described above. It requires checking whether zero, one, or two triangles exist, making it the most complex scenario for the law of sines.

Real-World Applications of the Law of Sines

Navigation and surveying are classic applications. If you know the distance between two landmarks and can measure angles to a third point, the law of sines gives you the distances to that third point. This technique, called triangulation, has been used since ancient times and remains fundamental to modern GPS and mapping.

In engineering, the law of sines helps analyze forces in trusses and structural frames where force vectors form triangles. In astronomy, it is used to calculate distances to stars using parallax measurements. In computer graphics, it helps with mesh calculations and ray-triangle intersection tests. Even in everyday problem-solving, such as determining the height of a building from two different positions, the law of sines provides an elegant solution.

Law of Sines vs Law of Cosines: When to Use Which

The law of sines requires at least one known side-angle pair (a side and its opposite angle). It works for AAS, ASA, and SSA configurations. The law of cosines does not require opposite pairs - it works with SAS (two sides and the included angle) and SSS (three sides).

When both laws could technically work, the law of sines is usually preferred because the math is simpler - it involves basic ratios rather than the quadratic-like formula of the law of cosines. However, for SAS and SSS problems, only the law of cosines works directly. A common strategy is to use the law of cosines to find one angle from SSS or SAS, then switch to the law of sines for the remaining calculations since it is simpler.