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 \(PABCD\)?

Guest Mar 27, 2019

#1**0 **

The volume of the pyramid is (1/3) (base area) height

The base area = AB * BC = 8 * 4 = 32

We can find the length of the base diagonal AC as

√ [ AB^2 + BC^2 ]= √ [ 8^2 + 4^2 ] = √ [ 64 + 16 ] = √80 = 4√5

And half of this distance is 2√5 = √20

So...the height = √[ PA^2 - (√200^2 ] = √ [ 6^2 - 20 ] = √ [ 36 - 20 ] = √ 16 = 4

So....the volume of the pyramid is

(1/3)(32)(4) =

(1/3) 128 =

128 / 3 units^3 = 42 + 2/3 units^3

CPhill Mar 27, 2019