# Ex.13.4 Q9 Surface Areas and Volumes Solution - NCERT Maths Class 9

## Question

A right circular cylinder just encloses a sphere of radius **\(r\)** (see **Fig. 13.22**).

Find:

(i) Surface area of the sphere

(ii) Curved surface area of the cylinder

(iii) Ratio of the areas obtained in **(i)** and** (ii)**.

## Text Solution

**Reasoning:**

The Curved surface area of the cylinder is given by \(\begin{align} = 2\pi rh \end{align}\) and Surface area of the sphere\(\begin{align}\, = 4\pi {r^2} \end{align}\)

**What is known?**

Radius of the sphere which touches the cylinder.

(i) Surface area of the sphere

**Steps:**

Radius of the sphere in \(= r\)

Surface area \( = 4\pi \rm{r^2} \)

(ii) Curved surface area of the cylinder.

**Steps:**

For cylinder: radius\(=r\)

Height \(=2r\)

Curved surface area of the cylinder

\[\begin{align}&= 2\pi rh \\&= 2 \times \pi \times r \times 2r \\&= 4\pi {r^2} \end{align}\]

(iii) Ratio of the areas obtained in **(i)** and **(ii)**.

\[\begin{align} &{\text{Ratio of the area}} \\ &\qquad = \frac{{{\text{Surface area of sphere}}}}{{\left( \begin{array}{l} {\text{Curved surface}} \\ {\text{area of cylinder}} \\ \end{array} \right)}} \\ &\qquad = \frac{{4\pi r^2 }}{{4\pi r^2 }} \\ &\qquad = \frac{1}{1} \\ &\qquad = 1:1 \\ \end{align}\]

**Answer:**

(i) Surface area of the sphere \(\begin{align} = 4\pi {r^2} \end{align}\)

(ii) Curved surface area of cylinder \(\begin{align} = 4\pi {r^2} \end{align}\)

(iii) Ratio between (i) and (ii) \(\begin{align} = 1:1 \end{align}\)