Area of a Triangle (AI HL)
Once a triangle isn't right-angled, plain SOHCAHTOA stops working and you need the sine and cosine rule toolkit instead. This page focuses on the cosine rule - for finding a missing side or angle from two sides and an included angle, or three known sides - and the sine-based area formula that goes with it. It's part of the broader Geometry & Trigonometry topic.
28 questions on this sub-topic.
The key formulas
Covered under IB syllabus reference SL3.2: the sine and cosine rules and the area of a triangle as \(\tfrac12 ab\sin C\) (this section excludes the ambiguous case). Both formulas below are in the formula booklet.
Area with sine
\[\text{Area}=\tfrac12 ab\sin C\]
Works for any triangle where two sides and the included angle are known - no height needed.
✓ In the formula bookletCosine rule
\(c^2=a^2+b^2-2ab\cos C\)
Finds a missing side from two sides and the included angle, or (rearranged) a missing angle from three known sides.
Need the full syllabus wording, the sine rule, and GDC guidance? See Geometry & Trigonometry (including using your GDC).
Worked examples
In triangle \(ABC\), \(b = 7\) cm, \(c = 10\) cm and \(A = 55^\circ\).
Find side \(a\).
Worked solution
Two sides and included angle \(A\): \(a^2=b^2+c^2-2bc\cos A\). M1
\(a^2=49+100-140\cos 55^\circ\approx 68.7\Rightarrow a\) A1
Two ships leave port \(P\) at the same time. Ship \(A\) travels 15 km on a bearing of \(040^\circ\) and ship \(B\) travels 22 km on a bearing of \(110^\circ\).
Find the distance \(AB\), to 3 significant figures.
Worked solution
The angle between the two bearings is \(110^\circ-40^\circ=70^\circ.\) A1
\(AB^2=15^2+22^2-2(15)(22)\cos70^\circ.\) M1
\(=225+484-660\cos70^\circ\approx 709-225.8=483.2\Rightarrow AB\approx 22.0\) km. A1
In quadrilateral \(ABCD\), \(AB = 10\), \(BC = 7\), angle \(ABC = 80^\circ\). Diagonal \(AC\) is found, then \(CD = 9\) and angle \(ACD = 40^\circ\).
(a) Find \(AC\).
(b) Find \(AD\).
Worked solution
(a) Find \(AC\). Cosine rule in \(\triangle ABC\): \(AC^2=100+49-140\cos 80^\circ\approx 124.7\Rightarrow AC\) M1
\(\approx 11.2.\) A1
(b) Find \(AD\). Cosine rule in \(\triangle ACD\), carrying the unrounded \(AC^2\approx124.69\) (not the rounded 11.2) to avoid compounding rounding error: \(AD^2=124.69+81-2\sqrt{124.69}(9)\cos 40^\circ\approx 51.7\Rightarrow AD\approx 7.19.\) M1A1A1
Common mistakes
- Reaching for SOHCAHTOA on a non-right-angled triangle. If none of the triangle's angles is \(90^\circ\), plain sine/cosine/tangent ratios don't apply - you need the sine or cosine rule instead.
- Misreading a bearings diagram. The angle to use in the cosine rule is the angle between the two paths at the shared point, not either bearing on its own - subtract the two bearings (or use the geometry of the diagram) to find it first.
- Forgetting a negative cosine still gives a valid angle. If \(\cos C\) comes out negative, \(C\) is obtuse - don't assume you've made an error, and don't force the answer to be acute.
Ready to practise properly?
28 area-of-a-triangle questions, marked instantly like the real exam.
Quick answers
What is the formula for the area of a triangle using sine?
\(\text{Area}=\tfrac12 ab\sin C\), using two sides and the angle between them - no height needed.
What is the cosine rule?
\(c^2=a^2+b^2-2ab\cos C\). Use it to find a missing side from two sides and the included angle, or a missing angle from three known sides.