Law of cosines: formula, calculator and worked example

What the symbols mean

$c$: the side opposite angle C
$a$: one side next to angle C
$b$: the other side next to angle C
$C$: the angle between sides a and b

When to use it

Use it when you know two sides and the angle between them (SAS) and want the third side, or when you know all three sides (SSS) and want an angle. Those are the two cases the law of sines cannot start. With a right angle, $cos 90° = 0$ and it becomes Pythagoras. Finding a side next to the known angle leads to a quadratic, which can have two positive answers (the ambiguous case again).

Worked example

Two sides of a triangle are 5 cm and 7 cm, and the angle between them is 60°. Find the third side.

Start from the formula.
$c^{2} = a^{2} + b^{2} - 2 a b \cos (C)$
Rearrange to make $c$ the subject: take the square root of both sides.
$c = \sqrt{a^{2} + b^{2} - 2 a b \cos (C)}$
Substitute $a$ = $5$ , $b$ = $7$ , $C$ = $60 °$ and work it out.
$c = \sqrt{5^{2} + 7^{2} - 2 \cdot 5 \cdot 7 \cos (60 °)}$
$= \sqrt{25 + 49 - 10 \cdot 7 \cdot 0.5}$
$= \sqrt{74 - 70 \cdot 0.5}$
$= \sqrt{74 - 35}$
$= \sqrt{39}$
$\approx 6.245$

Answer c ≈ 6.24 cm

Why it works

Drop a perpendicular from the far corner onto side b and apply Pythagoras to the two right triangles; the $−2abcos C$ term is the correction for the angle not being 90°. It is positive when C is obtuse, making c longer than Pythagoras would say.

Common mistakes

Computing $(a² + b² − 2ab)cos C$ . Only the last term is multiplied by cos C.
Using an angle that is not between sides a and b.
Forgetting the square root after finding $c^{2}$ .
Three sides that break the triangle inequality (one longer than the other two together) give no triangle; the inverse cosine then has no answer.

Law of cosines

Calculator

See it move

What the symbols mean

When to use it

Worked example

Why it works

Common mistakes