Law of Sines Calculator

Introduction

The Law of Sines Calculator helps you solve triangles quickly and easily. The Law of Sines is a rule in geometry that shows how the sides of a triangle relate to its angles. It says that when you divide any side of a triangle by the sine of the angle across from it, you always get the same number. This works for every pair of sides and angles in the triangle. You can use this calculator to find a missing side or a missing angle when you already know at least one side and its opposite angle. Just enter the values you have, and the calculator does the math for you. It is useful for homework, classwork, or any time you need to solve a triangle that is not a right triangle.

How to use our Law of Sines Calculator

Enter the angles and sides of a triangle that you already know, and this calculator will find all missing angles, sides, and properties using the Law of Sines. It also shows step-by-step work, a diagram, and the formula used.

Choose a mode: Select "Solve Full Triangle" to find all missing parts at once, "Find a Missing Side" to solve for one unknown side, or "Find a Missing Angle" to solve for one unknown angle. Each mode adjusts the inputs to guide you toward the right combination of values.

Select a case type (Full Triangle mode): Pick "ASA / AAS" if you know two angles and one side, or pick "SSA (Ambiguous)" if you know two sides and one angle opposite one of those sides. The SSA case may give you zero, one, or two valid triangles.

Enter Angle A, Angle B, and/or Angle C: Type the values of any known angles into the corresponding fields. Angles must be greater than zero and less than 180° (or π radians). If you provide two angles, the calculator will find the third automatically.

Enter Side a, Side b, and/or Side c: Type the lengths of any known sides into the matching fields. Side a is opposite Angle A, side b is opposite Angle B, and side c is opposite Angle C. All side lengths must be positive numbers.

Angle Unit: Choose whether your angles are in degrees or radians. The calculator will use the same unit for all angle inputs and outputs.

Length Label: Pick a unit of measurement for the sides, such as cm, m, ft, or in. This label is added to your results for clarity but does not change the math.

Precision: Select how many digits you want in your answers. "Smart" mode picks a clean level of detail automatically, or you can choose a specific number of significant figures or decimal places.

Show options: Use the checkboxes to turn the diagram, step-by-step solution, triangle properties, and formula display on or off based on what you need.

Click "Solve Triangle": Press the button or hit Enter to run the calculation. The calculator will display all solved angles and sides, the Law of Sines formula with your values substituted in, a labeled triangle diagram, a full step-by-step breakdown, and extra properties like area, perimeter, altitudes, medians, inradius, and circumradius.

Law of Sines Calculator

The Law of Sines is a rule in geometry that shows the relationship between the sides of a triangle and the angles across from them. It says that if you divide any side of a triangle by the sine of the angle opposite that side, you always get the same number. Written as a formula, it looks like this:

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

Here, a, b, and c are the three sides of the triangle, and A, B, and C are the angles opposite those sides. This rule works for every triangle, not just right triangles. For problems that specifically involve right triangles, you may find our Right Triangle Calculator more convenient.

When Do You Use the Law of Sines?

The Law of Sines is most helpful when you know some parts of a triangle and need to find the rest. There are two common situations where it applies:

• ASA or AAS (two angles and one side): If you know two angles, you can find the third angle since all three angles in a triangle always add up to 180°. Then you can use the Law of Sines to find the missing sides.
• SSA (two sides and one non-included angle): If you know two sides and an angle opposite one of them, you can use the Law of Sines to find the missing angle. This case is sometimes called the ambiguous case because it can produce zero, one, or two valid triangles.

When you have two sides and the included angle (SAS) or all three sides (SSS), the Law of Cosines Calculator is the appropriate tool to use instead.

The Ambiguous Case (SSA)

The SSA case needs special attention. When you know two sides and an angle opposite one of them, the math might give you two possible answers. This happens because the sine function gives the same value for an angle and its supplement (for example, sin(30°) = sin(150°) = 0.5). So there could be two different triangles that fit the same measurements. There could also be exactly one solution or no solution at all, depending on the sizes of the sides and angle.

• No solution: The side opposite the known angle is too short to form a triangle.
• One solution: Exactly one valid triangle can be formed.
• Two solutions: Two different valid triangles can be formed with the same given information.

How to Solve a Triangle Using the Law of Sines

Follow these basic steps:

1. Find the third angle if you already know two. Subtract the two known angles from 180°.
2. Set up the Law of Sines ratio. Pair a known side with its opposite angle, then set it equal to the unknown side paired with its opposite angle. Working with ratios is central to this method.
3. Solve for the unknown using cross-multiplication or basic algebra.

For example, if you know angle A = 40°, angle B = 60°, and side a = 10, you first find angle C = 180° − 40° − 60° = 80°. Then you use b = a · sin(B) / sin(A) to find side b, and c = a · sin(C) / sin(A) to find side c.

Useful Triangle Properties

Once all three sides and angles are known, you can also calculate other helpful measurements:

• Area: Found using the formula Area = ½ · b · c · sin(A). You can also explore our dedicated Triangle Area Calculator for other area methods.
• Perimeter: The sum of all three sides.
• Altitudes: The height from each vertex down to the opposite side.
• Medians: The line from each vertex to the midpoint of the opposite side.
• Inradius: The radius of the largest circle that fits inside the triangle. You can find the area of that inscribed circle from the inradius.
• Circumradius: The radius of the circle that passes through all three vertices, calculated as R = a / (2 · sin(A)).

The Law of Sines is one of the most important tools in trigonometry and geometry. It is used in fields like engineering, navigation, surveying, astronomy, and physics whenever you need to find unknown distances or angles in a triangle without a right angle. For finding straight-line distances between two points, our Distance Calculator can also be helpful, and if you need to determine the steepness of a line connecting two points, the Slope Calculator is a useful companion tool.

Frequently Asked Questions

What is the Law of Sines?

The Law of Sines is a math rule that says for any triangle, dividing a side by the sine of the angle across from it always gives the same number. The formula is: a/sin(A) = b/sin(B) = c/sin(C). It works for all triangles, not just right triangles.

What inputs do I need to use this calculator?

You need at least one side and two other pieces of information. The most common combos are:

• Two angles and one side (ASA or AAS)
• Two sides and one angle opposite a known side (SSA)

The calculator will figure out everything else from there.

What does ASA, AAS, and SSA mean?

ASA means Angle-Side-Angle — you know two angles and the side between them. AAS means Angle-Angle-Side — you know two angles and a side not between them. SSA means Side-Side-Angle — you know two sides and an angle opposite one of them. ASA and AAS always give one answer. SSA can give zero, one, or two answers.

What is the ambiguous case and why does it matter?

The ambiguous case happens with SSA (two sides and one non-included angle). Because sin of an angle equals sin of its supplement (for example, sin 30° = sin 150°), the given values might create two different valid triangles, one triangle, or no triangle at all. This calculator checks all possibilities and shows each solution if there are two.

Can I use this calculator with radians instead of degrees?

Yes. Use the Angle Unit dropdown to switch between degrees and radians. The calculator will use your choice for all inputs and outputs.

What is the difference between the three modes?

Solve Full Triangle finds every missing angle and side at once. Find a Missing Side guides you to solve for just one unknown side. Find a Missing Angle guides you to solve for just one unknown angle. The guided modes disable fields you don't need so you know exactly what to enter.

Why is a field grayed out or disabled?

In the guided modes (Find a Missing Side or Find a Missing Angle), the calculator disables fields you don't need for your chosen calculation. This helps you enter only the required values and avoids confusion.

What does the length label setting do?

The length label adds a unit name (like cm, ft, or m) to your results. It does not change the math. It just makes the output easier to read and understand.

What does the precision setting control?

Precision controls how many digits show in your answers. Smart picks a clean level automatically. You can also choose a set number of significant figures (3 to 6) or decimal places (2 to 4) if you need a specific level of detail.

What extra properties does the calculator show?

After solving the triangle, the calculator can show the area, perimeter, semi-perimeter, inradius, circumradius, all three altitudes, and all three medians. It also tells you the triangle type, such as acute scalene or obtuse isosceles. You can turn these on or off with the Show Properties checkbox.

What if the calculator says no solution exists?

This means the values you entered cannot form a real triangle. Common reasons include angles that add up to more than 180°, a side that is too short to reach the other side in the SSA case, or a sine value greater than 1. Double-check your inputs and try again.

Can I use the Law of Sines on a right triangle?

Yes, the Law of Sines works on all triangles, including right triangles. However, right triangles are often easier to solve using basic trigonometry (SOH-CAH-TOA) or the Pythagorean theorem.

How is the area of the triangle calculated?

The calculator uses the formula Area = ½ × b × c × sin(A). This works for any triangle once you know two sides and the angle between them. Since the calculator solves all sides and angles first, it always has the values needed for this formula.

What is the circumradius?

The circumradius is the radius of the circle that passes through all three corners of the triangle. The calculator finds it using the formula R = a / (2 × sin(A)).

What is the inradius?

The inradius is the radius of the largest circle that fits inside the triangle and touches all three sides. It is calculated as r = Area / s, where s is the semi-perimeter (half the perimeter).

When should I use the Law of Cosines instead?

Use the Law of Cosines when you know two sides and the included angle (SAS) or all three sides (SSS). The Law of Sines does not work well for those cases because you need a known side-angle pair to set up the ratio.