A regular triangular pyramid is cut by 2 planes which are parallel to its base. The planes trisects the altitude of the pyramid. Volume of top, middle and bottom part is V1, V2 and V3 respectively. What is the value of V1 :V2 : V3?

Solution:

We know that,

Volume of regular triangular pyramid ∝ side² × height and Side ∝ height

⇒ Volume of pyramid ∝ height³ = k height³

⇒ Volume of top pyramid = k (h/3)³ = kh³/27

⇒ Volume of middle portion = Volume of pyramid with second plane as base – Volume of top pyramid = {k(2h/3)³} – (kh³/27) = 7kh³/27

⇒ Volume of bottom portion = Volume of whole pyramid – Volume of second pyramid = kh³ – (8kh³/27) = 19Kh³/27

∴ Required ratio = 1 : 7 : 19