Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\overline{PA}\perp \overline{AB}$, $\overline{PA}\perp\overline{AD}$, and $PA = 6$, then what is the volume of $PABCD$?
To find the volume of the pyramid, I need to multiply the area of the base by the height and divide by 3. The base is a rectangle with length $AB = 8$ and width $BC = 4$, so its area is $8\times 4 = 32$. The height of the pyramid is the distance from the vertex $P$ to the base $ABCD$. Since $\overline{PA}$ is perpendicular to two adjacent edges of the base, it must also be perpendicular to the plane of the base, and therefore it is the height of the pyramid. So the height is $PA = 6$. Therefore, the volume of the pyramid is $\frac{1}{3}\times 32\times 6 = 64$. 