Tag: circle measures

Questions Related to circle measures

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 $\displaystyle cm^{2}$

Show answer
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} $

Show answer
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. $\dfrac{\pi}{6}$

Show answer
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. $15.75$

Show answer
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:

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