Law of Sines Calculator

What Is the Law of Sines and What Does It Help You Find?

The Law of Sines is one of the core triangle-solving rules in trigonometry. It connects each side of a triangle to the sine of the angle directly opposite that side. In a triangle labeled so that side a sits opposite angle A, side b sits opposite B, and side c sits opposite C, the rule is:

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

This relationship is incredibly useful because it turns triangle geometry into a proportional “matching game.” If you know one correct opposite pair (a with A, b with B, or c with C), you can often compute missing sides and angles using simple ratios. That’s why the Law of Sines shows up in surveying, navigation, physics, architecture, and any situation where a triangle models distances or directions you can’t measure directly.

How Do You Label a Triangle Correctly?

Many mistakes with the Law of Sines come from mixing up labels. The rule is not about “any side” and “any angle”—it is specifically about opposites. The easiest way to stay consistent is to remember: side a is always opposite angle A, and the same for b/B and c/C. If you are working from a diagram that uses different letters, rewrite the labels before you start, or map their labels to a/b/c and A/B/C so that every side-angle pair stays opposite.

When Should You Use the Law of Sines?

The Law of Sines works best when your known information includes an opposite side-angle pair and at least one additional value. In triangle shorthand: ASA (angle–side–angle), AAS (angle–angle–side), and SSA (side–side–angle where the angle is not included). It is usually not the first choice for SAS or SSS, where the Law of Cosines is the standard tool.

If you’re wondering “what if I only know angles,” that’s the AAA case. AAA tells you the triangle’s shape but not its size. You can draw infinitely many similar triangles with the same angles, so you need at least one side length to pin down real distances.

What Is the “k” Ratio and Why Does It Make Solving Easier?

A practical way to use the Law of Sines is to compute a single shared scale value: k = a / sin(A) (or b/sin(B), or c/sin(C) using whichever opposite pair you know). Once k is known, missing sides are straightforward: side = k · sin(angle).

This calculator shows k because it helps you sanity-check results. If your inputs are consistent, all three ratios a/sin(A), b/sin(B), and c/sin(C) should match (within rounding). If they don’t match, it’s a sign of a labeling error, an impossible triangle, or incorrect measurements.

How Does the Calculator Solve a Triangle?

Internally, the solver follows the same logic you would use by hand:

• It looks for at least one known opposite pair to compute the ratio k.
• It uses k to compute missing sides from known angles, or missing angles from known sides (via arcsin).
• It enforces the triangle angle-sum rule: A + B + C = 180° (or π radians).
• It checks for “domain issues” (for example, when side/k would require arcsin of a value greater than 1).

The key complication is the SSA ambiguous case, because arcsin can correspond to two valid angles in a triangle. That’s why the calculator may return two solutions.

Why Can SSA Create Two Solutions?

The sine function has a symmetry: sin(θ) = sin(180° − θ) (and similarly in radians). That means when you compute an angle with arcsin, you typically get the “principal” angle, but a second angle may also share the same sine value. In a triangle, either angle might fit depending on whether the remaining angle sum stays below 180° and whether the side lengths are consistent.

In real-world problems, the context often tells you which solution is correct. For example, a physical construction layout might eliminate an obtuse angle, or a navigation bearing might restrict a triangle to a particular orientation.

What Does “No Solution” Mean in Triangle Solving?

“No solution” doesn’t mean the calculator is broken—it means the numbers you entered cannot form a real triangle with the Law of Sines constraints. Common causes include:

• The angles add up to 180° or more before you even solve for the last angle.
• A computed ratio requires arcsin(x) where x > 1 (impossible in real numbers).
• You entered an opposite pair that doesn’t match the labeling (for example, using side a with angle B).
• Rounding or unit mix-ups (degrees vs radians) changed the meaning of an angle.

Degrees vs Radians: Which One Should You Choose?

Degrees are the default for most geometry classes, drafting, and everyday measurements. Radians are common in calculus, physics, and many engineering formulas. The calculator lets you choose either so you can match your source material exactly.

A quick self-check: if you enter an angle like 30 and you mean 30 degrees, you should not be in radians. In radians, 30 is enormous (far larger than π), and it will immediately break triangle logic.

Step-by-Step Example: ASA Triangle

Suppose you know A = 30°, B = 45°, and side a = 10. This is an ASA/AAS-style setup because you have two angles and one side. First, find the third angle: C = 180° − 30° − 45° = 105°. Then compute k = a/sin(A) = 10/sin(30°) = 10/0.5 = 20. Now find b and c: b = 20·sin(45°), c = 20·sin(105°). The result is a fully determined triangle with a unique solution.

Step-by-Step Example: SSA With Two Possible Solutions

In SSA, you might know A and side a, plus another side like b. You can compute k = a/sin(A), then compute B = arcsin(b/k). If B is acute, the “mirror” angle B' = 180° − B might also work, but only if A + B' is still less than 180°. When both are valid, you get two different triangles with different third angles and different remaining sides.

How to Avoid Common Mistakes

The Law of Sines is conceptually simple, but small input mistakes can cause big confusion. These checks help:

• Opposites must match: a with A, b with B, c with C.
• Angles must be valid: each angle must be greater than 0 and the sum must be less than 180° before solving the final angle.
• Units must match: do not mix degrees and radians in the same problem.
• SSA needs context: if two solutions appear, use real-world constraints to choose the correct one.

What If You Need More Than Just Sides and Angles?

Once your triangle is solved, you can compute many other quantities: perimeter (a+b+c), area (for example, (1/2)ab·sin(C)), and heights or medians. The Law of Sines is often the first stage in a broader geometry workflow: solve the triangle, then compute what your real problem asks for.

Where Is the Law of Sines Used in Real Life?

Triangles are the language of indirect measurement. Surveyors use triangles to compute distances across obstacles. Navigation uses triangles to convert bearings and measured legs into positions. In engineering, triangles model forces and components; in physics, they model wave directions and rotational relationships. Any time you can measure an angle and one distance but can’t measure another distance directly, the Law of Sines may help you compute what you need.

Why Does This Calculator Show Multiple Solutions Instead of Picking One?

In an SSA scenario, both solutions can be mathematically valid. Picking one automatically could be misleading if the “other” triangle is the one you actually intended. By showing 0/1/2 solutions clearly, the calculator helps you make an informed choice using your problem’s context (diagram, physical constraints, or wording).

How Accurate Are the Results?

Results are computed using standard floating-point math and then formatted to your chosen precision. If you need more digits for engineering or scientific work, increase the precision. If you’re reporting measurements from real-world instruments, remember that input uncertainty limits output accuracy—more decimals cannot create more real-world precision than your measurements contain.