Law of Sines Calculator

Use the Sine Rule to solve for unknown sides and angles in a triangle. Select the combination of values you know.

Enter the known values for your triangle to calculate the missing properties.

Practical Examples

Load an example to see how the calculator works with different inputs.

AAS (Angle-Angle-Side)

aas

Given two angles and a non-included side.

ASA (Angle-Side-Angle)

asa

Given two angles and the included side.

SSA (One Solution)

ssa_single

Given two sides and a non-included angle, resulting in one triangle.

SSA (Two Solutions)

ssa_ambiguous

An ambiguous case where two different triangles can be formed.

Other Titles
Understanding the Law of Sines: A Comprehensive Guide
Master the Sine Rule for solving triangles, from basic cases to the ambiguous SSA scenario.

What is the Law of Sines?

  • The core formula and its meaning
  • When to use the Law of Sines vs. Law of Cosines
  • The relationship between sides and their opposite angles
The Law of Sines (or Sine Rule) is a fundamental theorem in trigonometry that describes the relationship between the lengths of the sides of a triangle and the sines of their corresponding opposite angles. For any triangle with sides a, b, c and opposite angles A, B, C, the law states:
a/sin(A) = b/sin(B) = c/sin(C)
This elegant ratio means that if you know certain combinations of sides and angles, you can solve for the unknown parts of the triangle. It is most commonly used when you have a known angle-side pair.
When to Use It
The Law of Sines is your go-to tool for triangles when you know: 1. Two angles and any side (AAS or ASA). 2. Two sides and a non-included angle (SSA), also known as the ambiguous case.

Key Applications

  • AAS: Known Angle A, Angle B, and Side a.
  • ASA: Known Angle A, Side c, and Angle B.

Step-by-Step Guide to Using the Calculator

  • Selecting the right configuration (ASA, AAS, SSA)
  • Entering your known values correctly
  • Interpreting the results, including area and perimeter
Our calculator simplifies the process. Here's how to use it effectively:
1. Select Your Configuration
Start by choosing the set of values you know from the dropdown menu: ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), or SSA (Side-Side-Angle). The input fields will adapt based on your choice.
2. Enter Known Values
Fill in the corresponding input fields with your triangle's measurements. Ensure angles are in degrees. The calculator requires three inputs to solve the triangle.
3. Calculate and Analyze Results
Click 'Calculate' to see the solution. The results will show the unknown sides and angles, along with the triangle's perimeter and area. For the SSA case, it will automatically detect if one or two solutions exist.

Usage Scenarios

  • For ASA, if you know Angle A=30°, Side c=10, Angle B=70°, you can find Angle C, Side a, and Side b.
  • For SSA, if you know Side a=7, Side b=9, Angle A=45°, the calculator will determine if a valid triangle can be formed.

Real-World Applications of the Law of Sines

  • How it's used in surveying and mapping
  • Applications in astronomy and navigation
  • Its role in engineering and physics
The Law of Sines is not just an academic exercise; it has numerous practical applications.
Surveying and Triangulation
Surveyors use triangulation to measure distances between points. By measuring a baseline distance and the angles to a remote point from both ends of the baseline, they can use the Law of Sines to calculate the distance to that point without physically measuring it.
Astronomy
Astronomers use a similar principle, called parallax, to measure the distance to nearby stars. They measure a star's apparent position from two different points in Earth's orbit and use the Law of Sines to calculate its distance.

Practical Fields

  • Calculating the height of a mountain by measuring angles from two different points.
  • Determining the path of a satellite or a ship using angular measurements.

The Ambiguous Case (SSA): A Deeper Dive

  • Why the SSA case can lead to 0, 1, or 2 triangles
  • The mathematical conditions for each scenario
  • How the calculator handles ambiguity automatically
The Side-Side-Angle (SSA) case is known as the 'ambiguous case' because the given information may not define a unique triangle. Depending on the values, you might get no solution, one solution, or two distinct solutions.
Conditions for Ambiguity
Let the known parts be side a, side b, and Angle A. The ambiguity arises when side a is shorter than side b (a < b). We compare side a to the triangle's height (h = b * sin(A)).
  • If a < h: No solution. The side is too short to form a triangle.
  • If a = h: One solution (a right triangle).
  • If h < a < b: Two solutions. The side can swing to form two different valid triangles.
  • If a ≥ b: One solution.
Our calculator performs these checks for you, presenting all valid solutions without requiring manual calculations.

SSA Scenarios

  • a=5, b=10, A=30° → a = h, so one solution.
  • a=7, b=10, A=30° → h < a < b, so two solutions.

Mathematical Derivation and Formulas

  • Deriving the Law of Sines from basic principles
  • The formula for a triangle's area using sine
  • Connection to the circumcircle of a triangle
The Law of Sines can be derived by considering the area of a triangle or by using properties of its circumcircle.
Area-Based Derivation
The area of any triangle can be expressed in multiple ways: Area = (1/2)absin(C) = (1/2)bcsin(A) = (1/2)ac*sin(B). If you divide the entire expression by (1/2)abc, you get sin(C)/c = sin(A)/a = sin(B)/b. Taking the reciprocal gives the Law of Sines.
Connection to the Circumcircle
The ratio in the Law of Sines is also equal to the diameter of the triangle's circumcircle (the unique circle that passes through all three vertices). a/sin(A) = 2R, where R is the circumradius.

Core Formulas

  • Area = (1/2) * side a * side b * sin(Angle C)
  • a/sin(A) = b/sin(B) = c/sin(C) = 2R