Area of a Triangle (AA HL)
When you know two sides of a triangle and the angle trapped between them, you don't need a perpendicular height to find the area - one short sine formula does the job directly. This page covers that formula, how it interacts with the sine and cosine rules, and the mistakes that cost marks. It's part of the broader Triangles & Circular Functions topic.
12 questions on this sub-topic.
The area formula
Covered under IB syllabus reference SL3.2, alongside the sine rule \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) and the cosine rule \(c^2=a^2+b^2-2ab\cos C\). The area formula below is in the formula booklet.
Area formula
\[\text{Area}=\tfrac12ab\sin C\]
Works for any triangle where two sides and the angle between them are known - not just right triangles.
✓ In the formula bookletSetting up the triangle
Draw the horizontal and vertical legs first, mark the given angle at the horizontal, then decide whether the sine rule, cosine rule or simple right-angle trigonometry applies.
Need the sine rule and cosine rule too? See Triangles & Circular Functions.
Worked examples
Find the area of a triangle with sides \(a=10,\ b=12\) and included angle \(30^\circ.\)
Worked solution
Area \(= \tfrac12 ab\sin C = \tfrac12(10)(12)\sin30^\circ\) M1
\(= 60\cdot0.5\) A1
\(= 30.\) A1
A triangle has area \(20\) cm\(^2\), and two sides \(6\) cm and \(9\) cm. Find the acute included angle.
Worked solution
\(20=\frac12(6)(9)\sin\theta\Rightarrow\sin\theta=\dfrac{40}{54}\approx0.741.\) M1
\(\theta\approx47.8^\circ.\) A1 A1
Common mistakes
- Reaching for base times height when it isn't given. If only two sides and the included angle are known, \(\tfrac12ab\sin C\) is the right tool - you don't need to construct or find a perpendicular height first.
- Rounding too early. Carry full calculator accuracy through the area calculation and only round the final answer - early rounding can shift the last significant figure.
- Using the wrong pair of sides. \(C\) must be the angle actually enclosed between the two sides \(a\) and \(b\) you substitute - not any angle that happens to be labelled in the diagram.
- Assuming the area formula also hands you a missing side or angle. \(\tfrac12ab\sin C\) only outputs an area - working back to a missing side or angle from that area needs its own rearrangement, not a shortcut through the same formula - set up a fresh equation with the unknown as the subject instead of trying to invert \(\tfrac12ab\sin C\) directly, then solve that equation using the usual algebraic steps for whichever quantity the question actually asks for.
Ready to practise properly?
12 triangle-area questions, marked instantly like the real exam.
Quick answers
What is the formula for the area of a triangle using two sides and an angle?
\(\text{Area}=\tfrac12ab\sin C\), where \(a\) and \(b\) are two sides and \(C\) is the angle between them. It is in the formula booklet.
Does the triangle need a right angle to use this formula?
No. It works for any triangle where two sides and the included angle are known, unlike \(\tfrac12\times\text{base}\times\text{height}\), which needs a perpendicular height.