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Area of a sector of a circle - class-X

Description: area of a sector of a circle
Number of Questions: 56
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Tags: trigonometry geometry maths area of plane figures measurements circle measures areas related to circles
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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. $\dfrac {\theta}{720}\times 2\pi R^{2}$

Show answer
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. $\dfrac{\pi r^2}{2}$

Show answer
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

Show answer
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. $87.52cm^2$

Show answer
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

Show answer
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. $314 m^{2}$

Show answer
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. $176\, cm^2$

Show answer
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} $

Show answer
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

Show answer
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

Show answer
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

Given, $\displaystyle A = \frac{S}{360}\times \pi r^2$
$A$ is the area of setor, $ S$ is the angle measure in degrees of the sector and $r$ is the radius of the circle. Find $r$ in terms of $A$ and $S$.

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

  1. $r=\dfrac{360A\pi}{S}$

  2. $r=\dfrac{360A}{S\pi}$

  3. $r=\sqrt{\dfrac{360A\pi}{S}}$

  4. $r=\sqrt{\dfrac{360A}{S\pi}}$

Show answer
Correct Option: D
Explanation:

To change the formula in terms of $A$ and $S$, Isolate $r^2$, we get the formula as
$\dfrac{A\times 360}{S\pi}=r^2$
Now taking square root on both the sides to get the value of $r$ in terms of $A$ and $S$.
$\therefore r=\sqrt{\dfrac{360A}{S\pi}}$

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

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

  1. $13.2\ m^{2}$

  2. $14.2\ m^{2}$

  3. $13.4\ m^{2}$

  4. $14.4\ m^{2}$

Show answer
Correct Option: A
Explanation:

Area of sector $=$ $\dfrac { \theta  }{ { 360 }^{ 0 } } \times \pi { r }^{ 2 }=\dfrac { { 42 }^{ 0 } }{ { 360 }^{ 0 } } \times \dfrac { 22 }{ 7 } \times 6\times 6=13.2{ m }^{ 2 }$

Area of a sector having radius 12 cm and arc length 21 cm is

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

  1. 126 $cm^2$

  2. 252 $cm^2$

  3. 33 $cm^2$

  4. 45 $cm^2$

Show answer
Correct Option: A
Explanation:

Arc Length : Perimeter = Area of Sector : Area of Circle

$21: 2\pi r = \; Area \; of \;  Sector : \pi r^2$

$21:24\pi = \; Area \; of \;  Sector :144\pi$

Area of Sector $= \dfrac{144 \pi *21}{24 \pi} = 126cm^2$

If the area and arc length of the sector of a circle are 60 $cm^2$ and 20 cm respectively, then the diameter 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. 6 cm

  2. 12 cm

  3. 24 cm

  4. 36 cm

Show answer
Correct Option: B
Explanation:

Arc length of the Circle  : Area of the Sector = Perimeter of the Circle :Area of the Circle

Let the radius of the circle be 'r'.

 

Hence, $20 : 60$=$ 2\pi r : \pi r^2$

$ 1:3 = 2: r$

$ r= 6 $ (Product of Means = Product of Extremes)

Therefore, $Diameter = 2r = 12cm$

The perimeter of a sector of a circle is 37cm. If its radius is 7cm, then its arc length is 

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

  1. 23 cm

  2. 5.29 cm

  3. 32 cm

  4. 259 cm

Show answer
Correct Option: A
Explanation:

Perimeter of the Sector =37cm

Then, Radius = 7cm

Now perimeter of the sector of the circle $ =$ Arc's length+ Radius+radius 
37$ =$ Arc's Length +7+7

Arc's Length$ =$ 37-14 $=$ 23cm

The length of a minute hand of a wall clock is $8.4\ cm$. Find the area swept by it in half an hour.

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

  1. $100\ cm^{2}$

  2. $110.88\ cm^{2}$

  3. $120\ cm^{2}$

  4. $130\ cm^{2}$

Show answer
Correct Option: B
Explanation:

We know that minute hand covers $180^{o}$ in half an hour, which is a semicircle, hence area is

 $\Rightarrow \dfrac{1}{2}(\pi)(r^{2})=0.5\times3.1428\times(8.4)^{2}=110.88 \,cm^{2}$

The area of a sector of angle p (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. $\displaystyle \frac{p}{360} \times 2 \pi R$

  2. $\displaystyle \frac{p}{180}\times \pi R^2$

  3. $\displaystyle \frac{p}{720} \times 2 \pi R$

  4. $\displaystyle \frac{p}{720} \times 2 \pi R^2$

Show answer
Correct Option: D
Explanation:

Area of a sector with angle $p = \dfrac{p}{360} \times \pi \times R^2$ ,which matches with option D.

Find the area of sector whose length is $30\ \pi$ cm and angles of the sector is $40^o$.

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

  1. $2125\ \pi $ sq. cm

  2. $2225\ \pi $ sq. cm

  3. $2025\ \pi $ sq. cm

  4. $2200\ \pi $ sq. cm

Show answer
Correct Option: C
Explanation:
As we know that,
$1° = \cfrac{\pi}{180}$

$\therefore 40° = \cfrac{\pi}{180} \times 40 = \cfrac{2 \pi}{9}$

Let $S$ be the length of the arc and $A$ be the area of the corresponding sector.

Given that length of arc $\left( S \right) = 30 \pi \; cm$

As we know,
$S = r \theta$

$\Rightarrow 30 \pi = r \left( \cfrac{ \pi}{9} \right)$

$\Rightarrow r = 135 \; cm$

$\therefore$ Area of corresponding seector $\left( A \right) = \cfrac{1}{2} {r}^{2} \theta$

$\Rightarrow A = \cfrac{1}{2} {\left( 135 \right)}^{2} \left( \cfrac{2 \pi}{9} \right) = 2025 \pi \; {cm}^{2}$

The crescent shaded in the diagram, is like that found on many flags. $PSR$ is an arc of a circle, centre $O$ and radius $24.0$ cm. Angle POR $=$ $48.2^{\circ}$.
$PQR$ is a semicircle on $PR$ as diameter, where $PR$ $=$ $19.6$ cm
$[\pi = 3.14] [\cos 24.1 = 0.91]$

The area of the crescent is

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

  1. $116.4$ cm$^2$

  2. $123.4$ cm$^2$

  3. $112.2$ cm$^2$

  4. $23.4$ cm$^2$

Show answer
Correct Option: B
Explanation:

Length PQR $= \displaystyle  \frac{1}{2} \times 2 \pi r = \frac{1}{2} \times 2 \times 3.14 \times 9.8 = 30.772 cm$
Length PSR $ = \displaystyle \frac{\theta}{360} \times 2 \pi = \frac{48.2}{360}\times 2 \pi r= \frac{48.2 }{360} \times 2 \times 3.14 \times24$
$= 20.1797 cm$
Perimeter of crescent $= 30.772 + 20. 1797 = 51$
Area of sector POR $= \displaystyle \frac{\theta}{360} \times \pi \times r^2 = \frac{48.2}{360} \times 3.14 \times 14^2$
$= 242.16 cm^2 = 242 cm^2$
In triangle POR height, $ h = 24 cos24.1 = 21.91 cm$
Area $= \displaystyle \frac{1}{2} bh = \frac{1}{2} \times 19.6 \times 21.91= 214.70 cm^2$
$\therefore $ Area of shape PSR remaining $= 242.16 - 214.70 = 27.46 cm^2$
Area of semicircle $= \displaystyle \frac{1}{2} \times \pi \times 9.8^2 = 150.859$
$\therefore$ Area of crescent $=150.859 - 27.46 = 123.4 cm^2$

If the area of a sector of a circle is $\dfrac{5}{18}$th of the area of that circle, then the central angle of the sector is 100. Is it true or false?

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

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

For $180^\circ$, we have $\cfrac{1}{2}$ of the total area.

Hence, for $\cfrac{5}{18}^{th}$ of the total area, we have $360 \times \cfrac{5}{18} = 100^\circ$

An arc AB of a circle subtends an angle x radians at the centre O of the circle. Given that the area of the sector AOB is equal to the square of the length of the arc AB, then the value of x?

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

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

  2. $\dfrac{1}{4}$

  3. $\dfrac{1}{5}$

  4. $\dfrac{1}{2}$

Show answer
Correct Option: D

A wire of length $20\ cm$ can be bent $n$ the form of a sector then its maximum area is 

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

  1. $15\ sq.cm$

  2. $25\ sq.cm$

  3. $5\ sq.cm$

  4. $none$

Show answer
Correct Option: A

ABC is a right angel triangle right angled at vertex A. A circle is drawn to touch sides AB and AC at points P and Q respectively such that other end points of diameters passing through P and Q lie on side BC. If AB = 6. then the area of circular sector which lies outside the triangle is :

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

  1. $\pi -2$

  2. $\pi -3$

  3. 4

  4. $\pi +2$

Show answer
Correct Option: A

The area of the sector of circle ${x}^{2}+{y}^{2}=16$ and the line $y=x$ in the first quadrant is 

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

  1. $8\pi sq.units$

  2. $\pi sq.units$

  3. $4\pi sq.units$

  4. $2\pi sq.units$

Show answer
Correct Option: A

The area of a sector whose perimeter is four times its radius (r units)is

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

  1. $\sqrt{r}\,sq.\,units$

  2. ${r}^{4}\,sq.\,units$

  3. ${r}^{2}\,sq.\,units$

  4. $\displaystyle \frac {{r}^{2}}{r}\,sq.\,units$

Show answer
Correct Option: A
The radius of a circle is $7 cm$, then area of the sector of this circle if the corresponding angle is $30^{\circ}$ is 
maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $12.83 \,cm^2$

  2. $11.83 \,cm^2$

  3. $12.25 \,cm^2$

  4. None of these

Show answer
Correct Option: A
Explanation:

Area of a sector of a circle of radius '$r$' and angle $ = \dfrac { \theta  }{ 360 } \pi {r}^{2}$
Hence, area of the sector of the circle of  radius $ 7 $ cm and angle $ = \dfrac { 30 }{ 360 } \times \dfrac { 22 }{ 7 } \times 7 \times 7 = 12.83 \ \text{cm}^{2} $

The radius of a circle is $7 cm$, then area of the sector of this circle if the corresponding angle is:$210^{\circ}$ is 

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

  1. $88.83 \,cm^2$

  2. $87.83 \,cm^2$

  3. $89.83 \,cm^2$

  4. $86.83 \,cm^2$

Show answer
Correct Option: C
Explanation:

Area of a sector of a circle of radius 'r' and angle  $ \theta = \dfrac { \theta  }{ 360 } \pi {r}^{2}$

Hence,area of the sector of the circle of  radius $ 7 $ cm and angle $ { 210 }^{

0 } = \dfrac { 210 }{ 360 } \times \dfrac { 22 }{ 7 } \times 7 \times

7\quad = 89.83  {cm}^{2} $


The area of a circle is 314 sq. cm and area of its minor sector is 31.4 sq. cm. Find the area of its major sector.

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

  1. 282.6c$m^2$

  2. 200.6c$m^2$

  3. 180.04c$m^2$

  4. 1220.09c$m^2$

Show answer
Correct Option: A
Explanation:

Given:
Area of circle = $314 $$cm^2$
Area of minor sector = $31.4 $$cm^2$
Area of major sector = Area of a circle - Area of minor sector
= $314 - 31.4 cm^2$
= $282.6$ $cm^2$

The radius of a circle is $3.5$ cm and area of the sector is $3.85$ $cm^2$. Find the length of the corresponding arc.

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

  1. $2.2cm$

  2. $4.2cm$

  3. $5.1cm$

  4. $6.2cm$

Show answer
Correct Option: A
Explanation:

Let the angle of centre made by the sector be $\theta$
Therefore,
Area of the sector=$\pi r^2\dfrac{\theta }{360}$
                        $=>3.85=\dfrac{\pi(3.5)^2\theta}{360}$


                        $=>\theta=\dfrac{3.8\times 360\times 7}{(3.5)^2\times 22}$
                        $=35.5$
                        $=36$
Thus length of the arc =$2\pi r\dfrac{\theta}{360}$
                                   =$2\times \dfrac{22}{7}\times 3.5\times \dfrac{36}{360}$
                                   =$2.2cm$

The minute hand of a clock is $8: cm$ long. Find the area swept by the minute hand between $8.30: a.m.$ and $9.05: a.m.$

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

  1. $\displaystyle 117\frac{1}{3}:cm^{2}$

  2. $\displaystyle 107\frac{1}{3}:cm^{2}$

  3. $\displaystyle 217\frac{1}{3}:cm^{2}$

  4. None of these

Show answer
Correct Option: A
Explanation:

Angle made the centre by each $5$ minutes =$\dfrac{360}{12}$
                                                              =$30^o$


Angle covered between $8.30$am to $9.5$a.m is $210^o$

Therefore,
$Area=\pi(8)^2\dfrac{210}{360}$
        $=\dfrac{22}{7}\times 8\times 8\times \dfrac{210}{360}$
        $=117\dfrac{1}{3} cm^2$

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

Show answer
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$\displaystyle ^{2}$

Show answer
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

Show answer
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

Show answer
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. $3.6 cm$

Show answer
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. $183.3cm^2$

Show answer
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. $\displaystyle 6.42cm^{2}$

Show answer
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

Show answer
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:

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

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. $9 \pi$

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

Consider a circle with unit radius. There are seven adjacent sectors, $S _{1}, S _{2}, S _{3} ...S _{7}$, in the circle such that their total area is $\dfrac {1}{8}$ of the area of the circle. Further, the area of the $j^{th}$ sector is twice that of the $(j - i)^{th}$ sector, for $j = 2, .... 7$. Find the area of the sector $S _{1}$

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

  1. $\dfrac {\pi}{1016}$

  2. $\dfrac {\pi}{986}$

  3. $\dfrac {\pi}{116}$

  4. None

Show answer
Correct Option: A

Find the area of a sector of a circle of radius $28$cm and central angle $45^0$.

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

  1. $616 cm^{2}$

  2. $308 cm^{2}$

  3. $508 cm^{2}$

  4. $154 cm^{2}$

Show answer
Correct Option: B
Explanation:

Radius of sector $=28 cm$

Control angle $=45^{ o }$
Area of sector $=\cfrac { \theta  }{ 360° } \times \pi { r }^{ 2 }$
$=\cfrac { 45° }{ 360° } \times \cfrac { 22 }{ 7 } \times 28\times 28\ =308\quad { cm }^{ 2 }$

If a sector of a circle of diameter 21 cm subtends an angle of $120^{\circ}$ at the centre, then what is its area ? 

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

  1. $115.5 \ cm^2$.

  2. $84 \ cm^2$.

  3. $85.5 \ cm^2$.

  4. $78 \ cm^2$.

Show answer
Correct Option: A
Explanation:

Area of sector = $\cfrac{120}{360} \times \pi \times (\cfrac{21}{2})^2$

Thus area = $\cfrac{1}{3} \times \cfrac{22}{7} \times \cfrac{441}{4} = 115.5 cm^2$

To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle $80^{\circ}$ to a distance of 16.5 km. The area of the sea over which the ships are warned is 190 $km^2$ (app.).

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

  1. True

  2. False

  3. Nither

  4. Either

Show answer
Correct Option: A

If the sector of a circle of diameter $14 cm$ subtends an angle of $30^{\circ}$ at the centre, then its area is

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

  1. $49 \pi$

  2. $\displaystyle \frac{49 \pi}{12}$

  3. $\displaystyle \frac{242}{3\pi}$

  4. $\displaystyle \frac{121}{3\pi}$

Show answer
Correct Option: B,D
Explanation:

Area of a sector $=\dfrac{\theta}{360^0} \times \pi r^2 =\dfrac{30}{360} \times \pi (7)^2 = \dfrac{49 \pi}{12}$


Also, 
$ \dfrac{121}{3\pi}=\dfrac{121 \times 7}{3 \times 22} = \dfrac{49 \times 22}{12 \times 7} = \dfrac{49 \pi}{12}$

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

Show answer
Correct Option: A
Explanation:

Area of sector formed from angle $\theta$ = $\frac{\theta}{260} \pi r^2$, where r is the radius of the circle
Now, if angle is 120, 150 then the ratio of area of sector will be:
= $\frac{\frac{120}{360} \pi r^2}{\frac{150}{360} \pi r^2}$
= $\frac{120}{150}$ = 4:5

The area of a sector of a circle of angle $\displaystyle 60^{\circ}$ is $\displaystyle \frac{66}{7}cm^{2}$ then the area of the corresponding major sector is

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

  1. $\displaystyle 14cm^{2}$

  2. $\displaystyle \frac{55}{7}cm^{2}$

  3. $\displaystyle \frac{110}{7}cm^{2}$

  4. $\displaystyle \frac{330}{7}cm^2$

Show answer
Correct Option: D
Explanation:

Area of a sector of a circle of radius 'r' and angle  $ \theta = \frac {

\theta  }{ 360 } \pi {r}^{2}$
Given, $ \frac { 60 }{ 360 } \times \frac {22}{7} \times {r}^{2} = \frac {66}{7}  {cm}^{2} $
$ {r}^{2} = 18 $
Now, area of sector with angle {300}^{o} $ = \frac

{ 300 }{ 360 } \times \frac {22}{7} \times {r}^{2} = \frac

{ 300 }{ 360 } \times \frac {22}{7} \times 18 = \frac {330}{7}  {cm}^{2} $




A Car has two wipers which do not cover mutual area. Length of each wiper is 25 cms and it makes angle of $\displaystyle 115^{\circ}$ while cleaning. The area of cleaning by the wiper in one movement will be-

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

  1. $\displaystyle \frac{152815}{126}cm^{2}$

  2. $\displaystyle \frac{185125}{128}cm^{2}$

  3. $\displaystyle \frac{215815}{126}cm^{2}$

  4. $\displaystyle \frac{158125}{126}cm^{2}$

Show answer
Correct Option: D
Explanation:

Required area $\displaystyle =2\times \frac{115^{\circ}}{360^{\circ}}\pi \left ( 25 \right )^{2}$
$\displaystyle =2\times \frac{115}{360}\times \frac{22}{7}\times 625$


$\displaystyle=\frac{158125}{126}cm^{2}$

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