Solve Following Triangles Calculator

This calculator helps you solve triangles using the Law of Sines and Law of Cosines. Whether you're a student studying geometry or a professional working on engineering problems, this tool provides quick and accurate solutions for any triangle where you know at least three pieces of information.

How to Use This Calculator

To solve a triangle, you need to know at least three pieces of information: two sides and an included angle, or two angles and a side. The calculator will determine the remaining sides and angles based on your inputs.

Step-by-Step Instructions

Enter the known values in the appropriate fields. You can input sides (a, b, c) and angles (A, B, C) in degrees.
Select the units for angles (degrees or radians).
Click the "Calculate" button to solve the triangle.
Review the results, which will show all sides and angles of the triangle.
Use the reset button to clear all inputs and start over.

Note: The calculator will automatically determine which formula to use (Law of Sines or Law of Cosines) based on your inputs. If you provide inconsistent values, the calculator will display an error message.

Formulas Used

The calculator uses two fundamental trigonometric principles to solve triangles:

Law of Sines

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

The Law of Sines relates the lengths of the sides of a triangle to the sines of its opposite angles. This formula is particularly useful when you know two angles and one side, or two sides and a non-included angle.

Law of Cosines

c² = a² + b² - 2ab cos(C)

b² = a² + c² - 2ac cos(B)

a² = b² + c² - 2bc cos(A)

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. This formula is essential when you know two sides and the included angle, or all three sides.

For best results, ensure your inputs are consistent with the triangle inequality theorem (the sum of any two sides must be greater than the third side).

Worked Examples

Let's look at a couple of examples to see how the calculator works in practice.

Example 1: Two Sides and Included Angle

Suppose you have a triangle with sides a = 5, b = 7, and included angle C = 30°. Here's how you would solve it:

Enter a = 5, b = 7, and C = 30° in the calculator.
Click "Calculate".
The calculator will use the Law of Cosines to find side c: c² = 5² + 7² - 2*5*7*cos(30°) = 25 + 49 - 35*0.866 ≈ 25 + 49 - 30.34 ≈ 43.66 → c ≈ 6.60
Then it will use the Law of Sines to find angles A and B.

Example 2: Two Angles and One Side

Suppose you have a triangle with angles A = 40°, B = 60°, and side a = 8. Here's how you would solve it:

Enter A = 40°, B = 60°, and a = 8 in the calculator.
Click "Calculate".
The calculator will first find angle C: C = 180° - 40° - 60° = 80°.
Then it will use the Law of Sines to find sides b and c: b = (a * sin(B)) / sin(A) ≈ (8 * sin(60°)) / sin(40°) ≈ (8 * 0.866) / 0.6428 ≈ 10.99 → b ≈ 11.0, c ≈ (8 * sin(80°)) / sin(40°) ≈ (8 * 0.9848) / 0.6428 ≈ 12.19 → c ≈ 12.2.

Remember that all angles in a triangle must add up to 180°, and the sum of any two sides must be greater than the third side.

Frequently Asked Questions

What is the difference between the Law of Sines and the Law of Cosines?

The Law of Sines relates the sides of a triangle to the sines of its opposite angles, while the Law of Cosines relates the sides to the cosine of one of its angles. The Law of Sines is generally used when you know two angles and a side, while the Law of Cosines is used when you know two sides and the included angle.

Can I solve a triangle with only two sides and a non-included angle?

Yes, you can use the Law of Sines to solve a triangle when you know two sides and a non-included angle. The calculator will determine the possible solutions (either one or two triangles can fit the given information).

What if I enter inconsistent values that don't form a valid triangle?

The calculator will check for triangle inequality (the sum of any two sides must be greater than the third side) and display an error message if the values don't form a valid triangle. Make sure your inputs satisfy these conditions.

Can I use radians instead of degrees?

Yes, the calculator accepts angle inputs in both degrees and radians. Simply select the appropriate unit from the dropdown menu before entering your values.