Question

A right circular cylinder just encloses a sphere of radius r (see given figure). Find

(i) surface area of the sphere,
(ii) curved surface area of the cylinder,
(iii) ratio of the areas obtained in (i) and (ii).

Answer 1

We are given a right circular cylinder that just encloses a sphere of radius \( r \). We need to find the following:

• (i) Surface area of the sphere
• (ii) Curved surface area of the cylinder
• (iii) Ratio of the areas obtained in (i) and (ii)

Step-by-Step Solution:

(i) Surface Area of the Sphere:

The formula for the surface area \( A_{\text{sphere}} \) of a sphere is: \[ A_{\text{sphere}} = 4 \pi r^2 \] where \( r \) is the radius of the sphere.

(ii) Curved Surface Area of the Cylinder:

The cylinder just encloses the sphere, meaning the radius of the base of the cylinder is also \( r \), and the height of the cylinder is twice the radius of the sphere (since the cylinder must enclose the entire sphere vertically). The formula for the curved surface area \( A_{\text{cylinder}} \) of the cylinder is: \[ A_{\text{cylinder}} = 2 \pi r h \] where \( r \) is the radius of the base of the cylinder and \( h \) is the height of the cylinder. Since the height of the cylinder is twice the radius of the sphere, we have: \[ h = 2r \] Substituting this into the formula for the curved surface area: \[ A_{\text{cylinder}} = 2 \pi r \times 2r = 4 \pi r^2 \]

(iii) Ratio of the Areas:

The ratio of the surface area of the sphere to the curved surface area of the cylinder is: \[ \text{Ratio} = \frac{A_{\text{sphere}}}{A_{\text{cylinder}}} = \frac{4 \pi r^2}{4 \pi r^2} = 1 \]

Final Answer:

• Surface area of the sphere = \( \boxed{4 \pi r^2} \)
• Curved surface area of the cylinder = \( \boxed{4 \pi r^2} \)
• Ratio of the areas = \( \boxed{1} \)