The Cosine Rule

When the sine rule can't start — because you don't have a matching side-and-angle pair — the cosine rule steps in.

It connects all three sides to one angle:

c² = a² + b² − 2ab·cos C

Here C is the angle, and c is the side opposite it (the one you usually want to find). The other two sides, a and b, sit either side of angle C.

See it balance — and become Pythagoras

Drag the corners. The left and right of the equation always stay equal. Now watch angle C: as it approaches 90°, cos C → 0, the whole −2ab·cos C term vanishes, and you're left with c² = a² + b² — Pythagoras! The cosine rule is just Pythagoras with a correction term for non-right angles.

Drag any corner ▲

When the angle at C reads 90°, look for the green flag — both sides of the equation match the Pythagoras version exactly.

When do you use it?

✓ Two sides + the angle between them

(SAS) — find the third side directly from the formula.

✓ All three sides

(SSS) — rearrange to find any angle: cos C = a² + b² − c²2ab

Rule of thumb. Got a side-and-its-opposite-angle pair? Start with the sine rule — it's less arithmetic. No such pair (just SAS or SSS)? Use the cosine rule.

Worked example

Two sides of a triangle are a = 7 cm and b = 10 cm, with an angle of C = 60° between them. Find side c.

c² = a² + b² − 2ab·cos C
c² = 7² + 10² − 2×7×10×cos 60°
c² = 49 + 100 − 140 × 0.5
c² = 149 − 70 = 79
c = √79 ≈ 8.9 cm

New to all this? Start with the interactive Sin, Cos & Tan demo to build the intuition first.