Tag: surface area of a cone

Questions Related to surface area of a cone

If the area of the base of a right circular cone is 51 $ \displaystyle m^{2}     $and volume is 68 $ \displaystyle m^{3}     $ then its vertical height is

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

  1. 3.5 m

  2. 4 m

  3. 4.5 m

  4. 5 m

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Correct Option: B
Explanation:

Given the area of  base of right circular cone is 51 sq m and volume is 68 cu m

Then area of base =$\pi r^{2}=51$
$\Rightarrow r^{2}=\frac{51}{\pi }$
$\therefore volume =\frac{1}{3}\pi r^{2}h=68$
$\Rightarrow\frac{1}{3} \pi \left ( \frac{51}{\pi } \right )h= 68$
$\Rightarrow h=\frac{68}{17}$
$\Rightarrow h=4$ m

The radii of two right circular cone are int eh ratio of 4 : 5 and their slant heights are in the ratio 2:3 Then the ratio of their curved surfaces is

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

  1. 6 : 17

  2. 8 : 15

  3. 1 : 1

  4. none of these

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Correct Option: B
Explanation:

Let the radii of two right circular cone is 4x and 5x and siant height is 2y and 3y

Then curved surface area of first  right circular cone=$\pi \times 4x\times 2y=8\pi xy$ sq cm
And curved surface area of second right circular cone=$\pi \times 5x\times 3y=15\pi xy$ sq cm
So ratio of both right circular cone=$8\pi xy:15\pi xy::8:15$

The circumference of the base of a $24$ m high conical tent is $44$ m Calculate the length of canvas used in making the tent if the width of the canvas is $2$ m

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

  1. $257$ m

  2. $275$ m

  3. $752$ m

  4. $285$ m

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Correct Option: B
Explanation:

Let the radius of the base be r meters Then
$2$$\displaystyle \pi r$$=44$ m
or $\displaystyle r=\frac{44}{2\pi }=\frac{44\times 7}{2\times 22}=7 m$
Let the slant height be l meters Then
$\displaystyle l=\sqrt{h^{2}+r^{2}}=\sqrt{7^{2}+24^{2}}=\sqrt{625}=25m$
Now, Surface area = $\displaystyle \pi rl=\frac{22}{7}\times 7\times 25=550 m^{2}$
$\displaystyle \therefore $ Area of canvas required $= 550$ m$\displaystyle ^{2}$
 Given that width of the canvas is $2$ m therefore
Length of canvas required = $\displaystyle \frac{550}{2}=275m$

The slant height of a cone is $ 40$  m, the radius of the base is  $12$ m. Find the curved surface area of a cone.

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

  1. $\displaystyle 1507.2{ \ mm }^{ 2 }$

  2. $\displaystyle 1507.2\ m$

  3. $\displaystyle 1507.2{\  m }^{ 2 }$

  4. $\displaystyle 1407.2{\  mm }^{ 2 }$

Show answer
Correct Option: C
Explanation:

Using the forrmula for Curved surface area of the cone$= \pi rs$

where, $\pi=3.14$
$radius(r)=12\ m$
$slant\ height(s)=40\ m$
$\therefore$ Curved surface area$=3.14 \times 12 \times 40=1507.2m^2$

The formula used for surface area of cone is  ($s$ denotes slant height.)

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

  1. $\displaystyle \pi r\left( r+s \right) $ sq.units

  2. $\displaystyle \pi r\left( 2r+s \right) $ sq.units

  3. $\displaystyle 2\pi r\left( r+s \right) $ sq.units

  4. $\displaystyle \pi r{ \left( r+s \right) }^{ 2 }$ sq.units

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Correct Option: A
Explanation:

Surface area of a cone $=$ $\displaystyle \pi r\left( r+s \right) $ sq.units
Since, surface area of cone $=$  Area of sector $+$ Area of circle.

The curved surface area of the cone is $\pi r l$ whereas, total surface area of the cone is

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

  1. $\pi r(r+l)$

  2. $\pi rl(r+l)$

  3. $2\pi rl(r-l)$

  4. $2\pi r^2l(r+l)$

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Correct Option: A
Explanation:

Surface area of cone $=$ CSA of cone + area of circular base

                                    $=$ $\pi rl+\pi { r }^{ 2 }\Rightarrow \pi r(l+r)$

If the circumference at the base of a right circular cone and the slant height are $120\pi$ $cm$ and $10cm$ respectively, then the curved surface area of the cone is equal to

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

  1. $1200\pi$ ${cm}^{2}$

  2. $600\pi$ ${cm}^{2}$

  3. $300\pi$ ${cm}^{2}$

  4. $600$ ${cm}^{2}$

Show answer
Correct Option: B
Explanation:

We have,

circumference $=120\pi\ cm$, slant height$(l)=10\ cm$
$\Rightarrow 2\pi r=120\pi$
$\Rightarrow 2r=120$
$\Rightarrow r=60\ cm$

We know that the curved surface area of the cone
$=\pi r l$

So,
$=\pi \times 60 \times 10$

$=600\pi\ cm^2$

Hence, this is the answer

Two right circular cones have equal radii. If their slant heights are in the ratio $4:3$, then their respective curved surface areas are in the ratio

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

  1. $16:9$

  2. $2:3$

  3. $4:3$

  4. $3:4$

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Correct Option: C
Explanation:

We have,

The radius of the two cone are equal

The ratio of the slant height $l _1:l _2=4:3$

We know that the curved surface area of the cone
$=\pi r l$

So, the required ratio

$=\dfrac{\pi r l _1}{\pi r l _2}$

$=\dfrac{l _1 }{l _2 }$

$=\dfrac{4}{3}=4:3$

Hence, this is the answer.

The curved surface area of a right circular cone of height $15$ cm and base diameter $16$ cm is __________.

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

  1. $60\pi \ \text{cm}^2$

  2. $68\pi \ \text{ cm}^2$

  3. $120\pi \ \text{cm}^2$

  4. $136\pi \ \text{cm}^2$

Show answer
Correct Option: D
Explanation:

The formula of curved surface area of a right circular cone $ =\pi \times r\times l $

$ \Rightarrow l=\sqrt { { (h }^{ 2 } } +{ r }^{ 2 }) $
$ \Rightarrow l=\sqrt { 8^{ 2 } } +{ 15 }^{ 2 }) $
$ \Rightarrow l=17 $ cm
Now substitute the value in above equation. we have
Curved surface area $ =\pi \times 8\times 17 $
$ \Rightarrow 136\pi \ { \text{cm} }^{ 2 } $
So, option D is the correct.

The area of the base of a cone is $616\, cm^2$ and its height is 48 cm. The total surface area of cone is 

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

  1. $2816\, cm^2$

  2. $2861\, cm^2$

  3. $2618\, cm^2$

  4. $2681 \, cm^2$

Show answer
Correct Option: A
Explanation:
Given the area of the base of  a cone is $616\ cm^2$.
If $r$ be the radius of the base of the cone then 
$\pi r^2=616$
or, $\dfrac{22}{7}\times r^2=616$
or, $r=14$.
Also, given height $(h)=48\ cm$.
Then slant height $(l)=\sqrt{48^2+14^2}=2\sqrt{625}=50\ cm$.
$\therefore$ total surface area of the cone $=\pi r(r+l)=\dfrac{22}{7}\times 14\times 64=2816\ cm^2$.