Area Formula (SAS)
The SAS area formula calculates the area of any triangle given two sides and the included angle: Area = (1/2)*a*b*sin(C).
Formula
\text{Area} = \frac{1}{2} ab \sin(C)
Definition
If you know two sides of a triangle and the angle between them, you can find the area using $\text{Area} = \frac{1}{2}ab\sin(C)$; this follows because the height $h$ from the vertex opposite $b$ equals $a\sin(C)$, and when $C = 90^\circ$, $\sin(90^\circ) = 1$ and the formula reduces to $\frac{1}{2}ab$. Equivalently, $\text{Area} = \frac{1}{2}|\vec{a} \times \vec{b}|$, the magnitude of the cross product of the side vectors, which in $\mathbb{R}^2$ equals $\frac{1}{2}|\det([a_1,b_1; a_2,b_2])|$, linking the SAS formula to linear algebra.
Example
A triangle with sides $8$ and $5$ and a $40^\circ$ angle between them has area $\approx \frac{1}{2}(8)(5)(0.643) \approx 12.86$ square units; one with sides $12$ and $9$ and included angle $120^\circ$ has area $= \frac{1}{2}(12)(9)(\sqrt{3}/2) = 27\sqrt{3} \approx 46.77$ square units. The circumradius $R = abc/(4 \cdot \text{Area})$, and Hero's formula $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$ with $s = (a+b+c)/2$ is equivalent to the SAS formula after substituting the law of cosines.
Key Insight
This formula works for any triangle as long as you know two sides and the angle between them, unlike the base-times-height formula, which is easiest for right triangles; since area depends on $\sin(C)$, two fixed sides enclose the maximum area when they meet at a right angle, useful in optimization problems. The SAS area formula is also the key to the law of sines: since $\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2}ac\sin(B) = \frac{1}{2}bc\sin(A)$, dividing each pair gives $a/\sin(A) = b/\sin(B) = c/\sin(C)$.