Area of a triangle with no right angle

#### Tier: #Higher

Description

For any triangle where you know two sides and the included angle, use the formula: $$\text{Area} = \frac{1}{2}ab\sin C$$

where $a$ and $b$ are two sides and $C$ is the angle between them.

Example: A triangle with sides $a = 7\text{ cm}$, $b = 9\text{ cm}$, and included angle $C = 40°$: $$\text{Area} = \frac{1}{2} \times 7 \times 9 \times \sin 40° = \frac{1}{2} \times 63 \times 0.643 \approx 20.2\text{ cm}^2$$

This formula is essential when the perpendicular height is not given and cannot easily be found. It is derived from the standard $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ using $\text{height} = b\sin C$.

Common error: using the formula with an angle that is not between the two given sides. $C$ must be the angle enclosed by sides $a$ and $b$.

Links

Substitution Sine rule Cosine rule

Enter $\frac{1}{2} \times a \times b \times \sin(C)$ in one go — use brackets if needed.