what is the ratio of the small cone to the whole cone after when the cone is cut into two - Maths - Surface Areas and Volumes

Question

what is the ratio of the small cone to the whole cone after when
the cone is cut into two parts at the middle of its height????

Lalit Mehra · Accepted Answer

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)

$\Rightarrow \frac{OA}{OC} = \frac{AB}{CD} (Corresponding sides are proportional) \Rightarrow \frac{H}{(\frac{H}{2})} = \frac{R}{r}$

⇒ R = 2r

$Volume of smaller cone = \frac{1}{3} π (CD)^{2} \times OC = \frac{1}{3} π (r)^{2} \times \frac{H}{2} cm = \frac{π r^{2} H}{6} cm$

$V o l u m e o f w h o l e c o n e = \frac{1}{3} π (R)^{2} H = \frac{1}{3} π (2 r)^{2} H = \frac{4}{3} π r^{2} H \frac{Volume of smaller cone}{Volume of whole cone} = \frac{(\frac{π r^{2} H}{6})}{(\frac{4 π r^{2} H}{3})} = \frac{1}{8}$

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

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what is the ratio of the small cone to the whole cone after when the cone is cut into two parts at the middle of its height????