Tag: area of sectors and segments

Questions Related to area of sectors and segments

Tick the correct answer in the following:
Area of a sector of angle $\theta$ (in degrees) of a circle with radius R is

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

  1. $\dfrac {\theta}{180}\times 2\pi R$

  2. $\dfrac {\theta}{180}\times \pi R^{2}$

  3. $\dfrac {\theta}{3600}\times 2\pi R$

  4. $\dfrac {\theta}{720}\times 2\pi R^{2}$

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Correct Option: D
Explanation:

Area of a sector with angle $p$ $=\dfrac{\theta}{360}\times\pi R^2$

$=\dfrac{\theta}{360\times2}\times\pi R^2\times2$

$=\dfrac{\theta}{720}\times2\pi R^2$

Hence, Option $D$ is correct

If the angle subtended by the arc of a sector at the center is $90$ degrees, then the area of the sector in square units is

maths areas related to circles area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $2\pi r^2$

  2. $4\pi r^2$

  3. $\dfrac{\pi r^2}{4}$

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

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Correct Option: C
Explanation:

Since the Central angle is $90^{\circ}$, it means it is a Quad-circle.

So the Area of this sector is $\dfrac{1}{4}$ th of the Circle's Area $= \dfrac{1}{4}* \pi r^2$

The perimeter of a sector of a circle is $56$ cms and the area of the circle is $64\pi$ sq. cms  Find the area of sector.

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

  1. $360cm^2$

  2. $160cm^2$

  3. $260cm^2$

  4. None of these

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Correct Option: B
Explanation:

Area $= \pi r^{2}=64\pi cm^{2}$  


$\Rightarrow r=8cm$ 


perimeter $=2r+r\theta $ 

perimeter of sector $=r(\theta +2)=56cm$ 

$\Rightarrow \theta =5rad$ 

Area of sector $=\dfrac{r^{2}\theta }{2}=\dfrac{64}{2}\times 5cm^{2}$

                        $=160cm^{2}$

In a circle with radius $5.7\ cm$, the perimeter of a sector is $27.2\ cm$. Find the area of this sector.

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

  1. $97.52cm^2$

  2. $57.52cm^2$

  3. $77.52cm^2$

  4. $87.52cm^2$

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Correct Option: C
Explanation:
$R=5.7 cm$
Perimeter = $R\theta =27.2 cm$
$\therefore R\theta = 27.2 cm$
$\theta = \left(\dfrac{27.2}{5.7}\right)^{c}$
$\therefore $ Area of sector $=\dfrac{1}{2}R^{2}\theta $
$=\dfrac{1}{2}\times (5.7)^{2}\times \dfrac{27.2}{5.7}$
$=\dfrac{5.7}{2}\times 27.2 cm^{2}$
$ = 5.7 \times 13.6 = 77.52 cm^{2}$

The angle of sector with area equal to one fifth of total area of whole circle 

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

  1. 72

  2. 80

  3. 60

  4. 45

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Correct Option: A
Explanation:

The area of circle is $\pi r^2$

The area of sector is $\dfrac 15\pi r^2$
The area of sector is given as $\dfrac{x}{360}\times \pi r^=\pi r^2\\dfrac x{360}=\dfrac 15\x=72$

A horse is tied to a pole fixed at one corner of a $50 m \times 50 m$ square field of grass by means of a $20 m$ long rope. What is the area to the nearest whole number of that part of the field which the horse can graze?

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

  1. $1256 m^{2}$

  2. $942 m^{2}$

  3. $628 m^{2}$

  4. $314 m^{2}$

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Correct Option: D
Explanation:

The area of the field in which the horse can graze is one fourth of the circle of radius $ 20  cm $
Area of a circle $ = \pi { r }^{ 2 } $
Hence, area of the field in which the horse can graze $ = \cfrac {1}{4} \times
\cfrac{22}{7} \times 20 \times 20 = 314  $ sq m

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. $168\, cm^2$

  2. $148\, cm^2$

  3. $154\, cm^2$

  4. $176\, cm^2$

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Correct Option: B
Explanation:

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

The area of a sector is 1/18th of the area of the circle The sectorial angle is

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

  1. $\displaystyle 18^{\circ} $

  2. $\displaystyle 36^{\circ} $

  3. $\displaystyle 10^{\circ} $

  4. $\displaystyle 20^{\circ} $

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Correct Option: D
Explanation:

$\displaystyle \frac{x}{360}=\frac{1}{18}\Rightarrow x=20^{\circ}$

The minute hand of a clock is $\displaystyle \sqrt{21}$ cm long. The area described by the minute hand on the face of the clock between $7$ am and $7.05$ am is

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

  1. $5.5$ $\displaystyle cm^{2}$

  2. $22$ $\displaystyle cm^{2}$

  3. $11$ $\displaystyle cm^{2}$

  4. None of these

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Correct Option: A
Explanation:
Given $r=\sqrt{21}cm $

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

Therefore, angle made in $5$ minutes $=$ ${ 6 }^{ 0 }\times 5={ 30 }^{ 0 }$

Area of the sector$=\dfrac{\theta}{360^\circ}\times \pi \times r^2$

Here $\theta=30^\circ$

Hence, area swept in $5$ minutes $=$ $\dfrac { { 30 }^{ 0 } }{ { 360 }^{ 0 } } \times \dfrac { 22 }{ 7 } \times { \left( \sqrt { 21 }  \right)  }^{ 2 }$

                                                      $=$ $\dfrac { 1 }{ 12 } \times \dfrac { 22 }{ 7 } \times 21$

                                                      $=$ $5.5$ ${ cm }^{ 2 }$

A circular disc of radius 10 cm is divided into sectors with angles $ \displaystyle 120^{\circ}   $ and  $ \displaystyle 150^{\circ}   $ then  the ratio of the areas of two sectors is

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

  1. 4 : 5

  2. 5 : 4

  3. 2 : 1

  4. 8 : 7

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Correct Option: A
Explanation:

Now $\frac{c}{circle}=\frac{120}{360}=\frac{1}{3}$

And $\frac{c}{circle}=\frac{150}{360}=\frac{5}{12}$
So sector with angle 120 and 150 is part $\frac{1}{3}$ and $\frac{5}{12}$
Now ratio of the area of two sectors =Ratio of central angle =120:150=4:5