Tag: area of a sector of a circle

Questions Related to area of a sector of a circle

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

Find the area of a sector in radians whose central angle is $45^o$ and radius is $2$.

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

  1. $\dfrac{\pi}{3}$

  2. $\dfrac{\pi}{4}$

  3. $\dfrac{\pi}{2}$

  4. $\dfrac{\pi}{6}$

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Correct Option: C
Explanation:

Given: $\theta = 25^o = \dfrac{\pi}{4}$
Sector area $=$ $\dfrac{\theta}{2}r^2$
$=$ $\dfrac{\frac{\pi}{4}}{2}\times 2^2$ $=$ $\dfrac{\pi}{2}$

Find the area of a sector with an arc length of $20 cm$ and a radius of $6 cm$.

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

  1. $20$ $cm^2$

  2. $40$ $cm^2$

  3. $60$ $cm^2$

  4. $80$ $cm^2$

Show answer
Correct Option: C
Explanation:

Area of sector $=$ $\dfrac { Arc.length }{ 2\pi r } \times \pi { r }^{ 2 }$


                         $=$ $\dfrac { 20 }{ 2\pi r } \times \pi \times 6\times 6=60{ cm }^{ 2 }$

The area of a sector with a radius of $2 cm$ is $12 $$cm^2$. Calculate the angle of the sector. 

(Assume $\pi = 3$)

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

  1. $360^o$

  2. $160^o$

  3. $90^o$

  4. $180^o$

Show answer
Correct Option: A
Explanation:
$r = 2$cm
$A = 12cm^2$
Area of sector $=\dfrac {\theta}{360} \times \pi r^2$

$12 = \dfrac {\theta}{360} \times 3 \times 2^2$

$\theta = \dfrac {12 \times 360}{3 \times 4}$

$\theta = 360^o$

What is the area of a sector with a central angle of $100$ degrees and a radius of $5$? (Use $\pi = 3.14$)

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

  1. $21.80$

  2. $11.56$

  3. $12.46$

  4. $15.75$

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Correct Option: A
Explanation:

Area = $\dfrac{n}{360}\pi r^2$
= $\dfrac{100}{360}\pi 5^2$
= $6.944\pi$
= 21.80

The area of a sector is $120\pi$ and the arc measure is $160^o$. What is the radius of the circle?

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

  1. $16.43$

  2. $11.43$

  3. $12.23$

  4. $10.43$

Show answer
Correct Option: A
Explanation:

$A _{sector}= \dfrac{n}{360}\pi r^2$
$120\pi= \dfrac{160}{360}\pi r^2$
$270=r^2$
$r = 16.43$

Points $A$ and $B$ lie on circle $O$ (not shown). $AO=3$ and $\angle AOB ={120}^{o}$. Find the area of minor sector $AOB$.

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

  1. $\dfrac{\pi}{3}$

  2. $\pi$

  3. $3 \pi$

  4. $9 \pi$

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Correct Option: C
Explanation:

Area of a sector is given by $\cfrac{\theta}{360} \times \pi \times r^2$ where $\theta$ is the angle made by the sector, $r$ is the radius of the circle.

Here, $\theta = 120^o$ and $r = 3$
$\therefore$ area of minor sector $= \cfrac{120}{360} \times \pi \times 9 = 3\pi$

The minute hand of a clock is $7\ cm$ long. Find the area traced by it on the clock face between $4{:}15$ p.m. and $4{:}35$ p.m.

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

  1. $59\ cm^{2}$

  2. $65\ cm^{2}$

  3. $51.3\ cm^{2}$

  4. $45\ cm^{2}$

Show answer
Correct Option: C
Explanation:

Time $= 20$ min

Angle made by minute hand in 1 minute  $=\dfrac { { 360 }^{ 0 } }{ { 60 }^{ 0 } } ={ 6 }^{ 0 }$

$\therefore $  In $20$ min  $={ 6 }^{ 0 }\times 20={ 120 }^{ 0 }$

$\therefore $  Area swept  $=\dfrac { \theta  }{ { 360 }^{ 0 } } \times \pi { r }^{ 2 }=\dfrac { { 120 }^{ 0 } }{ { 360 }^{ 0 } } \times \dfrac { 22 }{ 7 } \times 7\times 7=\dfrac { 154 }{ 3 } =51.3{ cm }^{ 2 }$