Law of Sines Calculator

Solve any triangle by providing a valid combination of sides and angles.

Enter at least one side and any two other values (sides or angles). The calculator will attempt to solve the triangle.

Results

What is the Law of Sines?

The Law of Sines is a fundamental theorem in trigonometry that establishes a relationship between the sides of a triangle and the sines of their opposite angles. For any triangle, regardless of whether it is a right-angled or an oblique triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This powerful rule is essential for solving triangles when you know certain combinations of sides and angles, a process known as triangulation. The solve triangles using the law of sines calculator is an indispensable tool for students, engineers, and scientists who need to find unknown triangle measurements quickly and accurately.

Law of Sines Formula and Explanation

The Law of Sines is expressed by the following formula.

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

This equation can also be flipped, which is often more convenient when solving for an unknown angle.

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

Variable Definitions

Variable	Meaning	Unit	Typical Range
a, b, c	The lengths of the sides of the triangle.	Unitless (e.g., cm, inches, meters)	Any positive number
A, B, C	The measures of the angles opposite to sides a, b, and c, respectively.	Degrees (°)	0° to 180°

This formula is applicable in scenarios where you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). Our solve triangles using the law of sines calculator efficiently handles all these cases.

Practical Examples

Example 1: Given Two Angles and a Side (AAS)

Imagine a scenario where surveyors need to determine the distance across a river. They measure two angles and the length of one side along the riverbank.

• Inputs: Angle A = 40°, Angle B = 60°, Side a = 100 meters
• First, find Angle C: C = 180° – 40° – 60° = 80°
• Using the Law of Sines:
• b / sin(60°) = 100 / sin(40°) ⇒ b = (100 * sin(60°)) / sin(40°) ≈ 134.7 meters
• c / sin(80°) = 100 / sin(40°) ⇒ c = (100 * sin(80°)) / sin(40°) ≈ 153.2 meters
• Results: Angle C = 80°, Side b ≈ 134.7m, Side c ≈ 153.2m.

Example 2: The Ambiguous Case (SSA)

This occurs when we are given two sides and a non-included angle, which can sometimes result in two possible triangles. Consider a triangle with the following properties:

• Inputs: Side a = 6, Side b = 8, Angle A = 35°
• Using the Law of Sines to find Angle B:
• sin(B) / 8 = sin(35°) / 6 ⇒ sin(B) = (8 * sin(35°)) / 6 ≈ 0.7648
• This gives two possible angles for B: B1 ≈ 49.9° and B2 = 180° – 49.9° = 130.1°.
• Case 1: Angle B1 = 49.9°. Angle C1 = 180° – 35° – 49.9° = 95.1°. Side c1 can be found using the Law of Sines.
• Case 2: Angle B2 = 130.1°. Angle C2 = 180° – 35° – 130.1° = 14.9°. Side c2 can also be found.
• The solve triangles using the law of sines calculator will alert you when such an ambiguous case arises.

How to Use This Law of Sines Calculator

Using our calculator is straightforward. Follow these steps to find the missing parts of your triangle:

1. Enter Known Values: Input the values for the sides and angles you know. You need at least three values, including at least one side.
2. Select Units: While this calculator assumes degrees for angles, remember that units for side lengths (cm, m, ft) should be consistent. The calculations are unitless.
3. Click Calculate: Press the “Calculate” button to process the inputs.
4. Interpret Results: The calculator will display the missing side lengths, angles, area, and perimeter. It will also show a visual diagram of the solved triangle and warn you if an ambiguous case with two possible solutions is detected.

Key Factors That Affect Law of Sines Calculations

• Input Accuracy: Small errors in input values, especially angles, can lead to significant inaccuracies in the results.
• AAS vs. ASA vs. SSA: The combination of known values determines the solution path. AAS and ASA cases yield a unique triangle.
• The Ambiguous Case (SSA): This is the most complex scenario. When given two sides and a non-included angle, there might be no solution, one solution, or two distinct solutions. Our solve triangles using the law of sines calculator is designed to identify and handle this ambiguity.
• Sum of Angles: The three angles in any triangle must sum to 180 degrees. The calculator validates this rule.
• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. The calculator checks if a valid triangle can be formed.
• Rounding: Using rounded intermediate values in manual calculations can compound errors. The calculator uses high-precision values throughout the process.

Frequently Asked Questions (FAQ)

1. When should I use the Law of Sines?

Use the Law of Sines when you know two angles and any side (AAS or ASA) or two sides and a non-included angle (SSA). For SSS or SAS cases, the Law of Cosines is the appropriate tool.

2. What is the ambiguous case of the Law of Sines?

The ambiguous case occurs with the SSA configuration, where the given information can construct two different valid triangles. This happens because an angle and its supplement (e.g., 50° and 130°) have the same sine value.

3. Can the Law of Sines be used for right triangles?

Yes, but it’s often simpler to use basic trigonometric ratios (SOH-CAH-TOA) or the Pythagorean theorem for right triangles.

4. Why does my input result in an error?

An error can occur if the inputs don’t form a valid triangle. For example, if the sum of two given angles is over 180°, or if a calculated sine value is greater than 1, no triangle is possible. The solve triangles using the law of sines calculator will notify you of such issues.

5. How are the area and perimeter calculated?

Once all sides and angles are known, the perimeter is simply the sum of the side lengths (a + b + c). The area is calculated using the formula: Area = 0.5 * a * b * sin(C).

6. Does the calculator handle different units?

The calculations are based on the numerical values you provide. The results will be in the same unit system as your input for side lengths (e.g., if you input sides in cm, the results will also be in cm).

7. What does it mean if there is “no solution”?

In an SSA case, if the side opposite the given angle is too short to reach the base, no triangle can be formed. For example, given Angle A = 40°, Side b = 10, and Side a = 5, side ‘a’ is not long enough to complete the triangle.

8. Can I solve a triangle with only three angles?

No, knowing only three angles is not enough to determine the side lengths. You can have an infinite number of similar triangles with the same angles but different side lengths. At least one side must be known.