Area of triangle and segment

Triangle

With angle $C$ between sides $a$ and $b$

\text{area of triangle} = \frac 12 ab \sin C

where $b \sin C$ is the altitude to base $a$ , and $a \sin C$ is the altitude to base $b$ .

Students should see the connection to $\text{area} = \frac 12 bh$ .

HL students should also see the connection to $\text{area} = \frac 12\lvert\bm a \times \bm b\rvert$ , from vectors.

A segment of a circle is a sector minus a triangle. $r$ is radius, $\theta$ is central angle in radians.

\begin{align*}\text{area of segment} &= \text{area of sector} - \text{area of triangle} \\ &= \frac 12 \theta r^2 - \frac 12 r^2 \sin \theta \\ &= \frac 12 r^2 (\theta - \sin\theta)\end{align*}