Tag: sector and arc of a circle

Questions Related to sector and arc of a circle

What is the length of an arc of a circle with a radius of $5$ if it subtends an angle of ${60}^{o}$ at the center?

maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

  1. $3.14$

  2. $5.24$

  3. $10.48$

  4. $2.62$

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Correct Option: B
Explanation:
Given:
Radius (r)$=5$
Angle $=60^o$

Arc length for a particular angle we can write as -
$=\dfrac{\theta}{360}\times (2\pi r)$

$=\dfrac{60}{360}\times 2\pi \times 5$

$=\dfrac{10\pi}{6}=5.24$

Option 'B'.

If the sector of a circle of diameter $10$ cm subtends an angle of $144^{\circ}$ at the centre, then the length of the arc of the sector is

maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

  1. $2\pi $ cm

  2. $4\pi $ cm

  3. $5\pi$ cm

  4. $6\pi $ cm

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Correct Option: B
Explanation:
Given, diameter $=10$ cm, $\theta=144^o$
Length of an arc of a circle $=\dfrac { \theta  }{ 360 } \times 2\pi { r }=\dfrac { 144 }{ 360 } \times 2\pi \times \dfrac { 10 }{ 2 } =4\pi $ cm 
Hence, option B is correct.

A circular wire of radius $7$ cm is cut and bend again into an arc of a circle of radius $12$ cm. The angle subtended by the arc at the centre is

maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

  1. $50^\circ$

  2. $210^\circ$

  3. $100^\circ$

  4. $60^\circ$

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Correct Option: B
Explanation:

Given, radius of circular wire $= 7$ cm

Circumference of wire $= 2 \pi r = 2 \pi (7) = 14 \pi$
Radius of arc $= 12$ cm

Angle subtended by the arc $= \dfrac{\text{arc}}{\text{radius}} = \dfrac{14 \pi}{12} = \dfrac{7 \pi}{6}$

Angle subtended by arc $=\cfrac{7\pi}{6}\times \cfrac{180}{\pi}= 210^{\circ}$

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

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:

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