A cone is cut into 3 parts by two cuts that are parallel to the base such that the heights of three parts are equal. Find the ratio of volume of three parts.

1 : 7 : 19

The cuts that are parallel to the base;

Δ AO₁B,  Δ AO₂C,  Δ AO₃D are similar triangles;

Therefore, ratio of height (h) = ratio of respective radius (r)

⇒ Volume of cone = \(\frac{1}{3}\) \(\pi \) r² h = \(\frac{1}{3}\) \(\pi \) h³

Now,

Ratio of volume of three triangles  Δ AO₁B,  Δ AO₂C  and  Δ AO₃D

= \(\frac{1}{3}\) \(\pi \) (AO₁)³ : \(\frac{1}{3}\) \(\pi \) (AO₂)³ : \(\frac{1}{3}\) \(\pi \) (AO₃)³

= (AO₁)³ : (AO₂)³ :  (AO₃)³

= (1R)³ : (2R)³ :  (3R)³

= 1 : 8 : 27

Ratio of volume of three parts = 1 : (8 -1) : (27 -8) = 1 : 7 : 19