General | Math & Conversions Law of Cosines Calculator Law of Cosines Calculator (Non-Right Triangles) The Law of Cosines is used to solve any general triangle (non-right triangle) when you know either all three sides (**SSS**) or two sides and the angle between them (**SAS**). Select your known case below. Select Your Known Case SSS (Side-Side-Side) SAS (Side-Angle-Side) Side a Side b Side c Angle $\alpha$ ($A$) Angle $\beta$ ($B$) Angle $\gamma$ ($C$) Solve Triangle Solved Triangle Sides Angles Step-by-Step Solution The Three Forms of the Law of Cosines The Law of Cosines is a powerful tool because it works for any triangle. It relates the three sides of a triangle to the cosine of one of its angles. Formulas to Find a Missing Side (SAS Case) If you know two sides and the included angle (SAS), use one of these formulas to find the third side: $$a^2 = b^2 + c^2 - 2bc \cos(\alpha)$$ $$b^2 = a^2 + c^2 - 2ac \cos(\beta)$$ $$c^2 = a^2 + b^2 - 2ab \cos(\gamma)$$ Formulas to Find a Missing Angle (SSS Case) If you know all three sides (SSS), you can rearrange the primary formula to find any angle: $$\alpha = \cos^{-1} \left( \frac{b^2 + c^2 - a^2}{2bc} \right)$$ $$\beta = \cos^{-1} \left( \frac{a^2 + c^2 - b^2}{2ac} \right)$$ $$\gamma = \cos^{-1} \left( \frac{a^2 + b^2 - c^2}{2ab} \right)$$ Law of Cosines vs. Law of Sines The Law of Cosines is typically the starting point for solving a triangle in the **SAS** and **SSS** cases. Once you have used the Law of Cosines to find one missing value, it is often easier and less ambiguous to use the **Law of Sines** to find the remaining angles. The Law of Sines (used in ASA, AAS, and SSA cases) is: $$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$$ The Law of Cosines avoids the "ambiguous case" (SSA), making it the safer choice when applicable. Frequently Asked Questions (FAQ) What is the Law of Cosines used for? The Law of Cosines is a fundamental rule in trigonometry used to find missing parts of any general triangle (non-right triangle). It is specifically used when you know two sides and the included angle (SAS) or all three sides (SSS). What is the Law of Cosines formula for finding side 'a'? The formula to find side 'a' when sides 'b' and 'c' and the included angle 'A' are known is: $a^2 = b^2 + c^2 - 2bc \cos(A)$. This formula is a generalization of the Pythagorean Theorem. When should I use the Law of Cosines instead of the Law of Sines? Use the Law of Cosines in two specific scenarios: \n1. **SSS Case:** When you know all three sides and need to find an angle.\n2. **SAS Case:** When you know two sides and the angle between them (the included angle) and need to find the opposite side. \nFor all other solvable cases (ASA, AAS), the Law of Sines is sufficient. Does the Law of Cosines work for right triangles? Yes. If $\gamma = 90^\circ$, then $\cos(\gamma) = 0$. The formula $c^2 = a^2 + b^2 - 2ab \cos(\gamma)$ simplifies to $c^2 = a^2 + b^2$, which is the Pythagorean Theorem. So, the Law of Cosines is a generalization that always works. Key Cosine Formulas Find Side 'a' (SAS) $$a^2 = b^2 + c^2 - 2bc \cos(\alpha)$$ Find Angle 'A' (SSS) $$\alpha = \cos^{-1} \left( \frac{b^2 + c^2 - a^2}{2bc} \right)$$ Angle Sum Rule $$\alpha + \beta + \gamma = 180^\circ$$ Related Triangle Tools Law of Sines Calculator Triangle Solver (General) Heron's Formula Calculator (Area) Right Triangle Calculator Degrees to Radians Converter