Solve a Triangle Calculator

Triangle Formulas:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] \[ c^2 = a^2 + b^2 - 2ab \cos C \]

Side a:

units

Side b:

units

Side c:

units

Angle A:

degrees

Angle B:

degrees

Angle C:

degrees

Results:

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1. What is the Triangle Solver?

The triangle solver calculates all unknown sides and angles of a triangle when given sufficient information (at least three elements including at least one side). It uses the Law of Sines and Law of Cosines to solve for missing elements.

2. How Does the Calculator Work?

The calculator uses two fundamental trigonometric formulas:

\[ \text{Law of Sines: } \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] \[ \text{Law of Cosines: } c^2 = a^2 + b^2 - 2ab \cos C \]

Where:

\( a, b, c \) — Lengths of the sides
\( A, B, C \) — Angles opposite the corresponding sides (in degrees)

Explanation: These formulas relate the sides and angles of any triangle, allowing calculation of missing elements when enough information is provided.

3. Triangle Solving Methods

Four Cases:

SSS (Side-Side-Side): When all three sides are known, use Law of Cosines to find angles.
SAS (Side-Angle-Side): When two sides and the included angle are known, use Law of Cosines to find the third side, then Law of Sines for remaining angles.
ASA (Angle-Side-Angle): When two angles and the included side are known, find the third angle and use Law of Sines for remaining sides.
SSA (Side-Side-Angle): When two sides and a non-included angle are known, may have 0, 1, or 2 solutions (ambiguous case).

4. Using the Calculator

Tips: Enter any three elements (including at least one side) to calculate the remaining elements. Angles must be in degrees (0-180). Sides must be positive values.

5. Frequently Asked Questions (FAQ)

Q1: What information do I need to solve a triangle?
A: You need at least three elements including at least one side (SSS, SAS, ASA, or SSA).

Q2: Why does SSA sometimes give two solutions?
A: The "ambiguous case" occurs when two different triangles can satisfy the given conditions (when the known angle is acute and the opposite side is shorter than the adjacent side).

Q3: Can I use this for right triangles?
A: Yes, but right triangles can be solved more simply using the Pythagorean theorem and basic trigonometric ratios.

Q4: What units should I use?
A: Any consistent units for sides (cm, inches, etc.). Angles must be in degrees.

Q5: What if I get an error message?
A: Check that you've entered valid values (positive sides, angles between 0-180°) and enough information (3 elements including a side). Some combinations may be impossible (like angles summing to more than 180°).