Methods for Solving Triangles
Four fundamental techniques underpin triangle calculations:
- Law of Cosines: Relates any angle to all three sides, allowing you to find missing sides or angles when you know mixed data.
- Law of Sines: Establishes proportionality between side lengths and the sines of their opposite angles, particularly useful for ambiguous cases.
- Angle Sum Property: The three interior angles of any triangle always total 180°, so knowing two angles immediately gives the third.
- Pythagorean Theorem: A special case of the law of cosines applied when one angle equals 90°.
Your approach depends on what information you start with: all three sides, a mix of sides and angles, or just two angles.
Law of Cosines for Finding Angles
When you know all three sides of a triangle, use the law of cosines to recover each angle. The formula is rearranged to isolate the cosine of each angle, which you then invert using the inverse cosine function (arccos).
cos(α) = (b² + c² − a²) ÷ (2bc)
cos(β) = (a² + c² − b²) ÷ (2ac)
cos(γ) = (a² + b² − c²) ÷ (2ab)
a, b, c— The three side lengths of the triangle.α, β, γ— The angles opposite to sides a, b, and c respectively.
Law of Cosines for Finding Sides
When you know two sides and the included angle between them, the law of cosines directly solves for the third side. This formula applies regardless of whether the triangle is acute, obtuse, or right-angled.
a = √(b² + c² − 2bc × cos(α))
b = √(a² + c² − 2ac × cos(β))
c = √(a² + b² − 2ab × cos(γ))
a, b, c— The three side lengths of the triangle.α, β, γ— The angles opposite to sides a, b, and c respectively.
How to Use This Calculator
The calculator offers multiple input pathways depending on your available data:
- Three sides known: Enter a, b, and c to retrieve all three angles.
- One angle and two adjacent sides: Provide the included angle (e.g., α) and the sides that form it (b and c) to find side a and the remaining angles β and γ.
- Two angles known: Input any two angles; the third is computed automatically from the angle sum property (180°).
For example, if you input angle α = 30°, side a = 4, and side b = 6, the calculator returns side c ≈ 7.84, angle β ≈ 48.59°, and angle γ ≈ 101.4°.
Common Pitfalls and Considerations
Avoid these mistakes when solving triangles.
- Angle Units Matter — Ensure your calculator is set to degrees or radians consistently. Inputting 30 degrees when the tool expects radians will produce wildly incorrect results. Always double-check the unit setting before solving.
- Triangle Inequality Constraint — The sum of any two sides must exceed the third side. If your sides are 3, 4, and 8, no triangle exists. The calculator will flag this, but verify your measurements are realistic before entering them.
- Ambiguous Case (SSA) — When you know two sides and a non-included angle (SSA), two valid triangles may exist. The law of sines can produce two possible values for a missing angle. Be aware that your result might have an alternative solution.
- Rounding Errors in Cascading Calculations — If solving step-by-step manually, avoid rounding intermediate results. Accumulated rounding error can distort the final answer. Use full precision throughout, then round only the final output.