$${\frac{dA}{dt} = \frac{1}{2} r \frac{r\,d\theta}{dt} = \frac{1}{2} r v_{\theta} = \frac{1}{2} r v\sin\alpha = \frac{1}{2} v r \sin\alpha = \frac{1}{2} v h = \frac{L}{2m}}$$