A triangle is a fundamental geometric shape with three sides and three angles. "Solving a triangle" means finding the lengths of all its sides and the measures of all its angles, given some initial information. This calculator helps you do just that, using the powerful principles of trigonometry.
The Basics of Triangle Solving
To uniquely define and solve a triangle, you generally need at least three pieces of information, and at least one of these must be a side length. The common cases for solving triangles are:
SSS (Side-Side-Side): All three side lengths are known.
SAS (Side-Angle-Side): Two side lengths and the included angle (the angle between them) are known.
ASA (Angle-Side-Angle): Two angles and the included side (the side between them) are known.
AAS (Angle-Angle-Side): Two angles and a non-included side are known.
SSA (Side-Side-Angle): Two side lengths and a non-included angle are known. This is often called the "ambiguous case" because it can sometimes lead to two possible triangles, one triangle, or no triangle at all. Our calculator aims to find one valid solution if it exists (typically the one with an acute angle for the unknown angle).
Key Trigonometric Laws Used
The calculator relies on two primary trigonometric laws:
The Law of Sines
The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides and angles in a given triangle. Mathematically, it's expressed as:
a / sin(A) = b / sin(B) = c / sin(C)
Where a, b, c are the side lengths and A, B, C are the angles opposite those sides, respectively.
This law is particularly useful when you know two angles and one side (ASA or AAS), or two sides and a non-included angle (SSA).
The Law of Cosines
The Law of Cosines is a generalization of the Pythagorean theorem. It relates the lengths of the sides of a triangle to the cosine of one of its angles. It has three forms:
c² = a² + b² - 2ab * cos(C)
a² = b² + c² - 2bc * cos(A)
b² = a² + c² - 2ac * cos(B)
This law is essential when you know all three sides (SSS) or two sides and the included angle (SAS).
How to Use This Calculator
Input Values: Enter the known side lengths (a, b, c) and/or angle measures (A, B, C) into the respective fields. Angles should always be in degrees.
Minimum Requirements: Ensure you provide at least three values, and at least one of them must be a side length.
Calculate: Click the "Solve Triangle" button.
View Results: The calculator will display all six triangle properties (sides and angles), along with the perimeter and area, if a unique solution can be found.
Error Messages: If the inputs are insufficient, contradictory, or form an impossible triangle, an error message will be displayed.
Clear: Use the "Clear" button to reset all input fields and results.
Example Calculation
Let's say you have a triangle where:
Side a = 5
Side b = 7
Angle C = 60°
This is an SAS (Side-Angle-Side) case. Here's how the calculator would solve it:
Find Angle A (Law of Sines): a / sin(A) = c / sin(C) 5 / sin(A) = 6.245 / sin(60°) sin(A) = (5 * sin(60°)) / 6.245 sin(A) = (5 * 0.866025) / 6.245 ≈ 0.6932 A = arcsin(0.6932) ≈ 43.86°
Find Angle B (Sum of Angles): A + B + C = 180° B = 180° - A - C B = 180° - 43.86° - 60° ≈ 76.14°
The calculator would then also compute the perimeter (5 + 7 + 6.245 = 18.245) and the area (0.5 * 5 * 7 * sin(60°) ≈ 15.155).
This Triangle Solver Calculator is a powerful tool for students, engineers, architects, and anyone needing to quickly and accurately determine the properties of a triangle based on limited information.