Law of Cosines: Find the Third Side of a Triangle

Calculates the remaining side from two sides and the angle between them (in degrees) with the law of cosines c = √(a² + b² − 2ab·cos C).

Explanation

The law of cosines gives you the third side of a triangle when you know two sides and the angle between them. It generalises the Pythagorean theorem, which only works for right triangles, to every triangle.

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

$a$ , $b$ — the two sides whose lengths you know
$C$ — the angle between them, entered in degrees
$c$ — the side you are after, the one opposite $C$

The pairing matters. $C$ has to be the included angle, sitting between $a$ and $b$ and facing $c$ . Feed in some other angle and you are solving a different triangle.

Example

Take $a = 5$ , $b = 7$ and $C = 60^\circ$ . Since $\cos 60^\circ = 0.5$ ,

c = \sqrt{5^2 + 7^2 - 2 \times 5 \times 7 \times 0.5} = \sqrt{25 + 49 - 35} = \sqrt{39}

so $c \approx 6.2450$ .

Notes

Angles go in as degrees, not radians.
When $C = 90^\circ$ the cosine term vanishes and the formula collapses to $c = \sqrt{a^2 + b^2}$ , the Pythagorean theorem.
An obtuse $C$ makes $\cos C$ negative, so $-2ab\cos C$ adds to the total and $c$ comes out longer than it would at a right angle. A sharper angle pulls $c$ shorter.
Keep $C$ strictly between $0^\circ$ and $180^\circ$ . Within that range the expression under the square root is always positive, so a real third side always exists.