Triangle Side Angle Calculator

This calculator relies heavily on the Law of Cosines, which connects the lengths of a triangle's sides to the cosine of one of its angles. For any triangle with sides \(a\), \(b\), and \(c\), and respective opposite angles \(\alpha\), \(\beta\), and \(\gamma\), the primary formula states that \(a^2 = b^2 + c^2 - 2bc \cos(\alpha)\). By rearranging this algebraic equation, you can solve directly for any unknown angle: \(\cos(\alpha) = \frac{b^2 + c^2 - a^2}{2bc}\). Conversely, if two sides and their included angle are known, the formula calculates the third side length via square root extraction. Additionally, the interior angles always sum to 180 degrees (\(\pi\) radians).

Imagine you have a triangular plot of land with side lengths \(a = 4\), \(b = 5\), and \(c = 6\). To find angle \(\alpha\) (opposite to side \(a\)), we substitute the values into the Law of Cosines rearrangement: \(\cos(\alpha) = \frac{5^2 + 6^2 - 4^2}{2 \times 5 \times 6}\). Simplifying the numerator gives \(25 + 36 - 16 = 45\), and the denominator gives \(60\). Thus, \(\cos(\alpha) = \frac{45}{60} = 0.75\). Taking the inverse cosine yields \(\alpha \approx 41.41^\circ\). Repeating this process for side \(b\) yields \(\cos(\beta) = \frac{4^2 + 6^2 - 5^2}{2 \times 4 \times 6} = \frac{27}{48} = 0.5625\), resulting in \(\beta \approx 55.77^\circ\). Finally, the third angle \(\gamma\) can be found using the angle sum rule: \(180^\circ - 41.41^\circ - 55.77^\circ = 82.82^\circ\).

Always verify your input values to ensure they form a valid triangle; for instance, the sum of any two side lengths must strictly exceed the length of the third side (the triangle inequality theorem). When working with angle inputs, double-check whether your calculator or drafting software is set to degrees or radians to avoid massive calculation errors.