Law of Sines Calculator

Category: Algebra II
- May 14, 2025
| |

The Law of Sines relates the sides of a triangle to the sines of the opposite angles: a/sin(A) = b/sin(B) = c/sin(C). Use this calculator to find missing sides or angles in any triangle.

Enter Triangle Values

c b a A B C

Angles

°
°
°

Sides

Enter at least 3 values (with at least one side) to solve the triangle.

Display Options

About the Law of Sines

The Law of Sines is a fundamental relationship in trigonometry that relates the sides of a triangle to the sines of the opposite angles.

When to Use the Law of Sines

The Law of Sines is useful when you have:

  • Two angles and any side (AAS or ASA)
  • Two sides and an angle opposite one of them (SSA, which may have 0, 1, or 2 solutions)
Ambiguous Case (SSA)

When you know two sides and an angle opposite one of them (SSA), there can be 0, 1, or 2 possible triangles that satisfy these conditions. This is known as the ambiguous case.

Area Using the Law of Sines
Area = 12 × a × b × sin(C)

This formula calculates the area of a triangle using two sides and the included angle.

Applications
  • Navigation and surveying
  • Physics and engineering
  • Astronomy
  • Geographic information systems (GIS)
Law of Sines Formula:
a / sin(A) = b / sin(B) = c / sin(C)

What Is the Law of Sines Calculator?

The Law of Sines Calculator helps you solve triangles quickly by finding missing sides or angles based on the Law of Sines. Whether you know two angles and a side (AAS or ASA), or two sides and an opposite angle (SSA), this tool calculates the unknown values efficiently.

It can also calculate the area of a triangle, helping users like students, engineers, and surveyors get accurate results without complicated manual calculations.

How to Use the Calculator

  • Choose your angle unit – Select between Degrees or Radians.
  • Input your known values – Fill in at least three values, with at least one being a side.
  • Set decimal places – Pick the desired precision for your results.
  • Choose additional options – Opt to display calculation steps or find the triangle’s area.
  • Click "Calculate" – View the missing sides, angles, area, and an optional diagram of the triangle.
  • Click "Reset" to clear inputs and start over.

Benefits of Using the Law of Sines Calculator

  • Quickly solve for missing triangle sides and angles.
  • Visualize the triangle layout and dimensions.
  • Choose between degrees or radians for flexibility.
  • See step-by-step calculation details for better understanding.
  • Find the area of a triangle easily.

This tool works hand in hand with Other Math resources like an Inverse Function Calculator to solve for inverses, a Logarithm Calculator for solving logarithms, or a Midpoint Calculator when working with coordinate Geometry.

Real-World Applications

  • Navigation – Solving for missing distances or angles between locations.
  • Surveying – Measuring land plots where direct measurement is challenging.
  • Engineering – Structural calculations for bridges, buildings, and supports.
  • Physics – Analyzing forces in non-right-angled systems.

Frequently Asked Questions (FAQ)

Do I need to know all three sides to use the calculator?

No, you only need at least three known values with at least one being a side.

Can I solve SSA triangles with this calculator?

Yes, the calculator handles the ambiguous case for SSA situations, even showing when two triangles are possible.

What if my angles don't add up to 180°?

The calculator checks for this and adjusts small errors automatically. However, if the discrepancy is too large, you’ll be prompted to correct your inputs.

How accurate are the results?

You can control the precision by selecting how many decimal places you want, making it ideal for both rough estimates and precise engineering needs.

Can I calculate the area of a triangle with this tool?

Yes, the area is calculated automatically if the necessary sides and angles are available.

Related Tools You Might Like

Whether you are solving triangles with this Law of Sines Calculator or finding the midpoint between two points with a Midpoint Calculator, these tools make mathematical problem-solving faster and easier.