Law of Cosines Calculator

What is a law of cosines calculator?

The law of cosines calculator solves a triangle when you know two of its sides and the angle between them (the “side-angle-side” case). You enter side $a$, side $b$, and the included angle $C$, and the calculator returns the length of the third side $c$ along with the two remaining angles $A$ and $B$.

The law of cosines is a generalization of the Pythagorean theorem. When the included angle is exactly $90°$ the cosine term vanishes and the formula collapses back to $c^2 = a^2 + b^2$, the familiar relation for a right triangle.

How does it work?

The third side comes directly from the law of cosines:

$c^2 = a^2 + b^2 - 2ab\cos C$

Taking the square root gives $c$:

$c = \sqrt{a^2 + b^2 - 2ab\cos C}$

Once all three sides are known, the angle opposite side $a$ is recovered by rearranging the same law:

$A = \arccos\left(\frac{b^2 + c^2 - a^2}{2bc}\right)$

Because the three interior angles of any triangle sum to $180°$, the last angle follows immediately:

$B = 180° - A - C$

The included angle $C$ must lie strictly between $0°$ and $180°$, and both given sides must be positive, for the triangle to exist.

Worked examples

Right triangle. With $a = 3$, $b = 4$, and $C = 90°$, the cosine term drops out, so $c = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$. The remaining angles are $A \approx 36.8699°$ and $B \approx 53.1301°$, recovering the classic 3-4-5 triangle.

Oblique triangle. With $a = 5$, $b = 7$, and $C = 60°$, we get $c = \sqrt{5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos 60°} = \sqrt{25 + 49 - 35} = \sqrt{39} \approx 6.2450$.

Practical notes

The law of cosines is most useful when the law of sines cannot start a solution — specifically in the side-angle-side and side-side-side cases, where no side and its opposite angle are known together. Surveyors, navigators, and engineers rely on it to compute distances across a baseline when only two legs and the angle between them can be measured.

If instead you know two angles and a side, or two sides and a non-included angle, the law of sines is the more direct tool. For the special right-triangle case you can also use the hypotenuse calculator, and to evaluate the cosine of the included angle on its own, see the trigonometry calculator.