Question

A solid right circular cone is cut into two parts at middle of it's height by a plane parallel to it's base. Find the ratio of the volume of the smaller cone to the whole cone

Answer 1

Let the height and the radius of whole cone be H  and R respectively.

The cone is divided into two parts by drawing a plane through the mid point of its height and parallel to the base. 

$oc = ca \: = \frac{h}{2} cm$
Let the radius of the smaller cone be r cm.

In ∆OCD and ∆OAB,

∠OCD = ∠OAB  (90°)

∠COD = ∠AOB  (Common)

∴∆OCD ∼ ∆OAB  (AA Similarity criterion)

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Thus, the ratio of smaller cone to whole cone is 1 : 8.

Answer 2

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Let the height and the radius of whole cone be H  and R respectively.

The cone is divided into two parts by drawing a plane through the mid point of its height and parallel to the base. 

Let the radius of the smaller cone be r cm.

In ∆OCD and ∆OAB,

∠OCD = ∠OAB  (90°)

∠COD = ∠AOB  (Common)

∴∆OCD ∼ ∆OAB  (AA Similarity criterion)

Thus, the ratio of smaller cone to whole cone is 1 : 8