1 A solid right circular cone is cut into two parts at the middle of its height by a - Maths - Surface Areas and Volumes

Question

1 A solid right circular cone is cut into two parts at the
middle of its height by a plane parallel to its base The ratio of
the volume of the small cone isto whole cone is : ?

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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1. A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base . The ratio of the volume of the small cone isto whole cone is : ?