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 \(PABCD\)?
+128475 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
