Tag: area of sectors and segments

Questions Related to area of sectors and segments

The area of the sector of a circle whose radius is 6 m when the angle at the centre is $\displaystyle 42^{\circ}$ is 

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $\displaystyle 13.2:m^{2}$

  2. $\displaystyle 14.2:m^{2}$

  3. $\displaystyle 13.4:m^{2}$

  4. $\displaystyle 14.4:m^{2}$

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Correct Option: A
Explanation:

Area of sector $\displaystyle =\frac{42}{360}\times \pi r^{2}$
$\displaystyle =\frac{42}{360}\times \frac{22}{7}\times6\times6=13.2m^{2}$

The area of a sector of a circle of radius $16$ cm cut off by an arc which is $18.5$ cm long is 

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $168$ cm$\displaystyle ^{2}$

  2. $148$ cm$\displaystyle ^{2}$

  3. $154$ cm$\displaystyle ^{2}$

  4. $176$ cm$\displaystyle ^{2}$

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Correct Option: B
Explanation:

$\displaystyle A=\frac{1}{2}l r=\frac{1}{2}\times 18.5\times 16=148cm^{2}$

A sector of $120^{\circ}$ cut out from a circle has an area of $9\displaystyle \frac {3}{7}$ sq cm. The radius of the circle is

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $3$ cm

  2. $2.5$ cm

  3. $3.5$ cm

  4. $3.6$ cm

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Correct Option: A
Explanation:

Given, area of an sector $=9\dfrac {3}{7}$ sq. cm , $\theta=120^0$
We know Area of sector $=\dfrac {\theta}{360}\times \pi r^2$
$\Rightarrow \displaystyle \frac {\theta}{360}\, \times\, \pi r^2\, =\, 9\, \displaystyle \frac {3}{7}$
$\Rightarrow \displaystyle \frac {120}{360}\, \times\, \displaystyle \frac {22}{7}\, \times\, r^2\, =\, \displaystyle \frac {66}{7}$
$\Rightarrow r^2\, =\, \displaystyle \frac {66}{7}\, \times\, \displaystyle \frac {360}{120}\, \times\, \displaystyle \frac {7}{22}\, =\, 9$
$\Rightarrow r\, =\, \sqrt 9\, =\, 3$ cm

The area of the sector of a circle, whose radius is $6$ m when the angle at the centre is $42^0$, is

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $13.2$ sq. m

  2. $14.2$ sq. m

  3. $13.4$ sq.m

  4. $14.4$ sq. m

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Correct Option: A
Explanation:
Given, $\theta=42^0$, radius $=6$ m
Area of sector $=\, \displaystyle \frac {\theta}{360}\, \times\, \pi r^2$
$=\displaystyle \frac {42}{360}\, \times\, \displaystyle \frac {22}{7}\, \times\, 6\, \times\,6$
$ =\, 13.2$ sq. m

A sector of $120^{\circ}$ cut out from a circle has an area of $9\displaystyle \frac{3}{7}$sq cm. The radius of the circle is

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $3 cm$

  2. $2.5 cm$

  3. $3.5 cm$

  4. $3.6 cm$

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Correct Option: A
Explanation:
Let radius of circle be $'r' cm$. Then,
$\cfrac { \theta  }{ 360° } \times \pi { r }^{ 2 }=9\cfrac { 3 }{ 7 } cm^2=\cfrac { 66 }{ 7 } cm^2$
$\Rightarrow \cfrac { 120° }{ 360° } \times \cfrac { 22 }{ 7 } \times { r }^{ 2 }=\cfrac { 66 }{ 7 } \Rightarrow { r }^{ 2 }=\cfrac { 66\times 7\times 360° }{ 120°\times 22\times 7 } =9$
$\Rightarrow r=\sqrt { 9 } =3 cm$

The minute hand of a clock is $10$ cm long. Find the area of the face of the clock described by the minute hand between $9$A.M and $9.35$A.M.

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $90.165cm^2$

  2. $112.6cm^2$

  3. $156.4cm^2$

  4. $183.3cm^2$

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Correct Option: D
Explanation:

We have,
Angle described by the minute hand in one minute $=6^o$


$\therefore$ Angle described by the minute hand in $35$ minutes $=(6\times 35)^o=210^o$

$\therefore$ Area swept by the minute hand in $35$ minutes.


$=$ Area of a sector of angle $210^o$ in a circle of radius $10$ cm

 $=\dfrac {\theta}{360} \pi {r _1}^2$

$= \dfrac{210}{360}\times \dfrac{22}{7}\times (10)^2\ cm^2$.....

$=183.3\ cm^2$

The minute hand of a clock is 7 cm long Find the area  traced out by the minute hand of the clock between 6 pm to 6:30 pm

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $\displaystyle 14.4cm^{2}$

  2. $\displaystyle 15.4cm^{2}$

  3. $\displaystyle 7.2cm^{2}$

  4. $\displaystyle 6.42cm^{2}$

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Correct Option: D
Explanation:

The total angle of $ 12 $ hours in a clock is 
$ { 360 }^{ 0 } $.
$ => 24 $ half hours $ = { 360 }^{ 0 } $.
This means for one half an hour, angle $ = \frac {{ 360 }^{ 0 }}{24} = { 15 }^{ 0 } $

Area of a sector of a circle of radius 'r' and angle 
$ \theta = \frac { \theta  }{ 360 } \pi {r}^{2}$
Hence, area of the sector of the circle of  radius $ 7 $ cm and angle $

{ 15 }^{ 0 } = \frac { 15 }{ 360 } \times \frac { 22 }{ 7 } \times 7 \times

7\quad = 6.42  {cm}^{2} $

A chord of a circle of radius 6 cm subtends an angle of $\displaystyle 60^{\circ}$ at the centre of the circle. The area of the minor segment is
(use $\displaystyle \pi =3.14$)

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. 6.54 $\displaystyle cm^{2}$

  2. 0.327 $\displaystyle cm^{2}$

  3. 7.25 $\displaystyle cm^{2}$

  4. 3.27 $\displaystyle cm^{2}$

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Correct Option: D
Explanation:

$\displaystyle \theta =60^{\circ}$, r = 6 cm
Area of minor segment = $\displaystyle \dfrac{36}{2}\left [ \dfrac{60\times 3.14}{180}-\dfrac{\sqrt{3}}{2} \right ]$
                                     = 3.27 $\displaystyle cm^{2}$

The area of a sector with  perimeter as  $45\ cm$ and radius as $6 \ cm$ is

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $44$ $ \displaystyle cm^{2} $

  2. $66$ $ \displaystyle cm^{2} $

  3. $88$ $ \displaystyle cm^{2} $

  4. $99$ $ \displaystyle cm^{2} $

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Correct Option: D
Explanation:

$\Rightarrow$  Perimeter of a sector $=45\,cm$


$\Rightarrow$  Radius of a circle $(r)=6\,cm$


$\Rightarrow$  Arc of sector $(l)=Perimeter\,of\,sector-2r$
                             $=45-(2\times r)$
                             $=45-(2\times 6)$
                             $=45-12$
                             $=33\,cm$

$\Rightarrow$  Area of sector $=\dfrac{1}{2}\times r\times  l\\$
                               $=\dfrac{1}{2}\times 6\times 33\\$
                               $=3\times 33$
                               $=99\,cm^2$

Arc of a sector is equal to-

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. Length of arc $\times$ radius

  2. $ \displaystyle \frac{sector angle}{360^{\circ}}\times circumference of circle $

  3. $ \displaystyle \frac{sector angle}{360^{\circ}}\times (area of circle) $

  4. None of these

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Correct Option: C