One angle of a parallelogram is 120 degrees, and two consecutive sides have lengths of 8 inches and 15 inches. What is the area of the parallelogram? Express your answer in simplest radical form.
I think we can use the formula for the area of a parallelogram, which is base times height. Yes, but we have to find the height. The height is the perpendicular distance from the base to the opposite side. Can we use the Pythagorean theorem to find the height? That's a great idea! Since one angle of the parallelogram is 120 degrees, the other must be 60 degrees. The side across from the $60^\circ$ angle is the height of the parallelogram. We can use the fact that $\sin(60^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}$ to find the height. So the height is $8\sin(60^\circ)$. So the area is base times height, which is $15 \cdot 8\sin(60^\circ)$. That simplifies to $120\sin(60^\circ)$. And since $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, the area is $120\cdot\frac{\sqrt{3}}{2}$, which is $60\sqrt{3}$.

