Tag: surface area of cone

Questions Related to surface area of cone

A cone and a cylinder have the same base area. They also have the same curved surface area. If the height of the cylinder is $3$ m, then the slant height of the cone (in m) is

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $3$

  2. $4$

  3. $6$

  4. $7$

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Correct Option: C
Explanation:

Given the radius of cylinder and cone are the same because there base areas are same.
Curved surface of cylinder $=$ curved surface area of the cone
$\therefore 2 \pi r h = \pi r l$
$\therefore l = 2h $
Given, height of cylinder $= 3$ cm
Therefore, slant height of cone $= 6$ cm

The curved surface area of a right circular cone with height $24$ cm and radius $7$ cm is

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $500 cm^2$

  2. $550 cm^2$

  3. $607 cm^2$

  4. $650 cm^2$

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Correct Option: B
Explanation:

Given,
Radius of cone $= 7$ cm
Height of cone $= 24$ cm
Curved surface area = $\pi r \sqrt{(r^2 + l^2)}$
= $\pi \times 7 (\sqrt{(7^2 + (24)^2)}$
= $\pi \times 70\times 25$
= $550 cm^2$

Consider two cones, the curved surface area of one being twice that of the other and the slant height of the later being twice that of the former. The ratio of the radius of the later cone to that of the former is

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $1 : 4$

  2. $1 : 2$

  3. $2 : 1$

  4. $4 : 1$

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Correct Option: A
Explanation:

Let the curved surface areas of two cones be $C _1$ and $C _2$

And their slant heights be $l _1$ and $l _2$
Given, $C _1=2C _2$
$l _2=2l _1$

$\therefore \pi r _1l _1=2\pi r _2.2l _1$
$\therefore r _1=4r _2$

The curved surface area of a right cone is $\displaystyle 286 m^{2}$ and its the slant height is 13 m, then volume is

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $\displaystyle 821.389m^{3}$

  2. $\displaystyle 852.258m^{3}$

  3. $\displaystyle 364.369m^{3}$

  4. $\displaystyle 281.164m^{3}$

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Correct Option: D
Explanation:

Curved surface area of a cone $= \pi rl$  where r is
the radius of the cone and l is the slant height.
Hence, CSA of this cone, $ = \frac {22}{7} \times r \times 13 = 286 $
$ => r = 7  m $

For a cone, l $ = \sqrt { { h }^{ 2 }+  {r}^{ 2 } } $ where lis the slant height.

Hence, $ 13 = \sqrt { { h }^{ 2 }+  {7}^{ 2 } } $ 

$ 169 = { h }^{ 2 } + 49 $

$ { h }^{ 2 } = 120 $

$ h = 2 \sqrt {30}  m $

Hence, volume of this cone $ = \frac { 1 }{ 3 } \times \frac { 22 }{ 7 } \times { 7 }^{ 2 }\times 2 \sqrt {30} = \frac {154}{3} \sqrt {30} { m }^{ 3 } $


The cost of the canvas required to make a conical tent of base radius $8$ m at the rate of Rs. $40$ per $\displaystyle m^{2}$ is Rs. $10,048$. Find the height of the tent .$\displaystyle \left ( Take\  \pi =3.14 \right )$ 

maths solids surface area of a cone surface area of cone the surface area of a cone

  1. $6$ m

  2. $7$ m

  3. $8$ m

  4. $10$ m

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Correct Option: A
Explanation:

To find the amount of canvas required to make a conical tent, we need to calculate the lateral surface area of the tent.
Lateral surface area of a cone of radius $ r $, height $ h = \pi r\sqrt{{h}^{2} + {r}^{2}} $
As the total cost to make the tent is Rs $10048 $ at the rate of Rs $ 40 $ per sq m, total area LSA $ = \dfrac {10048}{40} = 251.2 $ sq m 

So, $ \pi r \times \sqrt{{h}^{2} + {r}^{2}} = 251.2 $
$ \Rightarrow  3.14 \times 8 \times \sqrt{{h}^{2} + {8}^{2}} = 251.2 $
$ \Rightarrow  \sqrt{{h}^{2} + {8}^{2}} = 10 $
$ \Rightarrow  {h}^{2} + 64 = 100 $
$ \Rightarrow  {h}^{2}  = 36 $
$ \Rightarrow  h = 6 $ m