Question

9. A sphere, a cylinder and a cone are of the same radius
and same height. Find the ratio of their curved surface
areas?​

Answer 1

Given:

• A sphere, a cylinder and a cone are of the same radius and same height.

To find:

• Ratio of their curved surface areas?

Solution:

Here,

• Height of sphere = Diameter
• Height of cone = 2 × radius
• Height of cylinder = 2 × radius

Then,

• Curved surface area of sphere = 4πr²
• Curved surface area of cylinder = 2πrh = 2πr(2r) = 4πr²
• Curved surface area of cone = πrl

where,

l = √(r)² + (h)² = √(r)² + (2r)²

= √r² + 4r² = √5r² = r√5

• Therefore, Curved surface area of cone = π √(5r²)

Now, Finding Ratio of curved surface area of sphere, cylinder and cone,

⇏ 4πr² : 2πrh : πrl

⇏ 4πr² : 4πr² : π √(5r²)

⇏ 4 : 4 : √5

∴ Hence, the ratio of their curved surface areas is 4 : 4 : √5.

Answer 2

Solution

Given,

radius of sphere = radius of cylinder = radius of cone = r

Since,

the height of sphere is its diameter , so the its height = 2r

And ,

height of sphere = height of cylinder = height of cone = 2r

Now,

ratio of curved surface area of sphere , cylinder and cone ;

$4\pi \: r ^{2} \ratio \: 2\pi \: rh \ratio \: \pi \: rl$

$= 4\pi \: r ^{2} \ratio \: 2\pi \: rh \ratio \: \pi r \sqrt{ {r}^{2} + {h}^{2} }$

$= 4\pi \: r ^{2} \ratio \: 2\pi \: r(2r) \ratio \: \pi \: r \sqrt{ {r}^{2} + {2r}^{2} }$

$= 4\pi \: r ^{2} \ratio \: 2\pi \: r(2r) \ratio \: \pi r \sqrt{ {r}^{2} + {2r}^{2} }$

$= 4\pi \: r ^{2} \ratio \: 4\pi \: r ^{2} \ratio \: \pi \: r \times r \sqrt{5r}$

$= 4\pi \: r ^{2} \ratio \: 4\pi \: r ^{2} \ratio \: \pi \: r \times r \sqrt{5r}$

$= 4\pi \: r ^{2} \ratio4\pi \: r ^{2} \ratio\pi \: r ^{2} \sqrt{5}$

$= 4 \ratio \: 4 \ratio \: \sqrt{5}$

The ratio of there CSA area is 4:4:√5 .