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Length of an arc - class-IX

Description: length of an arc
Number of Questions: 24
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Tags: maths areas related to circles area of plane figures measurements circle measures geometry trigonometry circumference and area of a circle
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A sector of a circle with sectorial angle of $\displaystyle 36^{\circ} $ has an area of 15.4 sq cm 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. $8.8 m$

  2. $4.4 m$

  3. $0.22 m$

  4. $0.44 m$

Show answer
Correct Option: D
Explanation:

$\displaystyle \frac{36}{360}\times \frac{22}{7}r^{2}=15.4\Rightarrow r^{2} =\frac{15.4\times 5\times 7}{11}=49$
$\displaystyle \Rightarrow r=7$
$\displaystyle C=2\pi r=2\times \frac{22}{7}\times 7=44cm$
$\displaystyle =0.44 m$

In a circle of radius 21 cm an arc subtends an angle of $\displaystyle 56^{\circ} $ at the centre of the circle. The length of the arc is

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

  1. $20.53$ cm

  2. $17.53$ cm

  3. $15.53$ cm

  4. $16.53$ cm

Show answer
Correct Option: A
Explanation:

$\displaystyle \theta =56^{\circ},r=21cm$
Length. of $\displaystyle AB=\dfrac{56^{\circ}}{360^{\circ}}\times 2\times \dfrac{22}{7}\times 21$
$\displaystyle =\dfrac{616}{30}=20.53cm$

What is the length of arc AB making angle of $126^0$ at center of radius $8$?

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

  1. $2.6\displaystyle \pi $

  2. $5.6\displaystyle \pi $

  3. $7.6\displaystyle \pi $

  4. $\displaystyle \frac{1}{2}\pi $

Show answer
Correct Option: B
Explanation:

Setting a proportion 
AB : OB : : 126 : 360
$\displaystyle \frac{\overline{AB}}{2\pi r}=\frac{126}{360}$
$\displaystyle \overline{AB}=\left ( \frac{126}{360} \right )\times 2\pi r$
$\displaystyle \overline{AB}=\left ( \frac{126}{360} \right )\times 2\pi \times 8$
$\displaystyle \overline{AB}=5.6\pi $

If the radius and arc length of a sector are 17 cm and 27 cm respectively, then the perimeter is

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

  1. 16 cm

  2. 61 cm

  3. 32 cm

  4. 80 cm

Show answer
Correct Option: B
Explanation:

Perimeter of the Sector = 2(Radius of the circle) + Arc Length 


$Perimeter = 2(17) + 27 = 2*17 +27 = 61cm$

If an arc of a circle of radius 14 cm subtends an angle of $60^{\circ}$ at the centre, then the length of the arc is $\displaystyle \frac{44}{3} cm$.

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

  1. True

  2. False

  3. Niether

  4. Either

Show answer
Correct Option: A

In a circle of radius 21 cm, an arc subtends an angle of $60^{\circ}$ at the centre the length of the arc is 22 cm.

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

  1. True

  2. False

  3. Neither

  4. Either

Show answer
Correct Option: A

The length of an arc of a sector of a circle of radius r units and of centre angle $\theta$ is $\displaystyle \frac{\theta}{360^{\circ}} \times \pi r^2$.

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

  1. True

  2. False

  3. Neither

  4. Either

Show answer
Correct Option: B

Length of an arc of a circle with radius $r$ and central angle $\theta$ is(angle in radians):

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

  1. $\dfrac{r\times \theta}{360^{o}}$

  2. $\dfrac{r\times \theta}{180^{o}}$

  3. $\dfrac{r\times \theta}{90^{o}}$

  4. $r\times \theta$

Show answer
Correct Option: D
Explanation:
Let $r$ be the radius of a circle and $\theta$ be the central angle
Length of an arc of the sector $=r\times \theta$
Hence, length of an arc of a circle $=r\times \theta$.

If $ABC$ is an are of a circle and $\angle ABC=135^{o}$, then the ratio of arc $ ABC$ to the circumference is:

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

  1. $1 :4$

  2. $3:4$

  3. $3:8$

  4. $1:2$

Show answer
Correct Option: A
The diameter of a circle is $10$ cm, then find the length of the arc, when the corresponding central angle is $180^{\circ}$.  $(\pi =3.14)$
maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

  1. $15.7$

  2. $16$

  3. $3.14$

  4. $18$

Show answer
Correct Option: A
Explanation:

Radius of the circle $ = \dfrac {\text{Diameter}}{2} = 5 $ cm 


Length of an arc subtending an angle $ \theta  = \dfrac { \theta  }{ 360 }

\times 2\pi R $, where $R$ is the radius of the circle. 

So, length of the arc $ = \dfrac {180}{360} \times 2 \times 3.14\times 5  = 15.7 $ cm

The diameter of a circle is $10$ cm, then find the length of the arc, when the corresponding central angle is $144^{\circ}$.$(\pi =3.14)$
maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

  1. $44$ cm

  2. $12.56$ cm

  3. $12$ cm

  4. $88$ cm

Show answer
Correct Option: B
Explanation:

Radius of the circle $ = \dfrac {Diameter}{2} = 5  cm $

Length of an arc subtending an angle $ \theta  = \dfrac { \theta  }{ 360 }

\times 2\pi R $ where R is the radius of the circle. 



So, length of the arc $ = \dfrac {144}{360} \times 2 \times 3.14 \times 5  = 12.56  cm $

A sector is cut from a circle of radius $21$ cm. The angle of the sector is $150^o$. Find the length of its arc and area.

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

  1. $27$ cm and $412.7cm^2$

  2. $36$ cm and $436.9cm^2$

  3. $45$ cm and $517.5cm^2$

  4. $55$ cm and $577.5cm^2$

Show answer
Correct Option: D
Explanation:

The length of arc $l$ and area $A$ of a sector of angle $\theta$ in a circle of radius $r$ are given by,


$l=\displaystyle\frac{\theta}{360^o}\times 2\pi r$

and $A=\displaystyle\frac{\theta}{360^o}\times \pi r^2$ respectively.

Here, $r=21$ cm and $\theta=150^0$


$\therefore l = \dfrac{150}{360}\times2\times\dfrac{22}7\times21 = 55$ cm

and 

$A = \dfrac{150}{36}\times\dfrac{22}7\times21^2 = \dfrac{1155}2 = 577.5\ {cm}^2$

A horse is tied to a post by a rope If the horse moves along a circular path always keep the rope tight and describes $88$ metres when it has traced out $\displaystyle 72^{\circ}$ at the center, then the length of rope is 

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

  1. $60 m$

  2. $65 m$

  3. $70 m$

  4. $72 m$

Show answer
Correct Option: C
Explanation:

$\text{Since we know that angle in radian }\theta =\dfrac{arc length}{radius}$
Here, $\theta=72^0=\dfrac{\pi}{180}\times72=\dfrac{2\pi}{5}$
and arc length=88 m


$\therefore \dfrac{2\pi}{5} =\dfrac{88}{radius}$
radius= $70 m$

The perimeter and area of a sector are $18\;cm$ and $20\;sq.\,cm$ respectively. Then the length of the arc is:

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

  1. $10\;cm\;or\;8\;cm$

  2. $10\;cm\;or\;5\;cm$

  3. $10\;cm\;or\;4\;cm$

  4. $20\;cm\;or\;2\;cm$

Show answer
Correct Option: A
Explanation:

$l+2r=18$
$\displaystyle\frac{lr}{2}=20$
$lr=40$
$l=\displaystyle\frac{40}{r}$
$18=\displaystyle\frac{40}{r}+2r$
$r^2-9r+20=0$
$(r-4)\;\;(r-5)=0$
$r=4$ or $r=5$
$l=8$ or $l=10\;cm$

A sector is cut off from a circle of radius $21$ cm The angle of the sector is $\displaystyle 120^{\circ} $ The length of its arc is [Take $\displaystyle \pi =\frac{22}{7} $]

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

  1. $40 cm$

  2. $44 cm$

  3. $35 cm$

  4. $28 cm$

Show answer
Correct Option: B
Explanation:
Given radius$(r)=21cm$ and the angle$(\theta)=120^\circ$

length of arc$=r\times \theta$

here $r=21cm,$ $\theta=120^\circ=\dfrac{120}{360}\times 2\pi$

length of arc = $\dfrac { \theta  }{ { 360 }^{ 0 } } \times 2\pi r=\dfrac { { 120 }^{ 0 } }{ { 360 }^{ 0 } } \times 2\times \dfrac { 22 }{ 7 } \times 21=44cm$

How long is the arc subtended by an angle of $\dfrac{2\pi}{3}$ radians on a circle of radius $12$ cm?

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. $6\pi$ cm

  4. $8\pi$ cm

Show answer
Correct Option: D
Explanation:

We know that length of subtended arc $=$ $\theta r$
Here, $\theta = \cfrac{2\pi}{3}$
$r = 12$ cm
So, $ s = \cfrac{2\pi}{3}\times 12$
$s = 8\pi$ cm

What is the area of a sector with an arc length of $120 cm $ and radius $4cm$?

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

  1. $120$ $cm^2$

  2. $240$ $cm^2$

  3. $260$ $cm^2$

  4. $180$ $cm^2$

Show answer
Correct Option: B
Explanation:
$r = 4$cm
$l = 120cm$
Area of sector $=\dfrac {l \times r}{2}$

$A = \dfrac {120 \times 4}{2}$

$A = 240 cm^2$

Given radius = $11 $ cm, area of the sector is $230 $ $cm^2$. Find the length of the arc $SR$.

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

  1. $40.56 cm$

  2. $41.81 cm$

  3. $43.61 cm$

  4. $46.12 cm$

Show answer
Correct Option: B
Explanation:

Area of sector = $\dfrac{RL}{2}$
$230=\dfrac{11L}{2}$
$L = 41.81 cm$

A horse is tethered to a stoke by a rope $30\ m$ long. If the horse moves along the circumference of a circle always keeping the rope tight then how far it will have gone when the rope has traced an angle of $105^{\circ}$?

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

  1. $50\ m$

  2. $55\ m$

  3. $60\ m$

  4. $65\ m$

Show answer
Correct Option: B
Explanation:

Distance covered  $=\dfrac { \theta  }{ 360 } \times 2\pi r=\dfrac { 105 }{ 360 } \times 2\times \dfrac { 22 }{ 7 } \times 30=55m$


A pizza parlor cuts its $14$-inch (diameter) pizzas into $8$ equal slices. What is the size (in square inches) of each slice?

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

  1. $5.5$

  2. $19.2$

  3. $44.1$

  4. $60.4$

  5. $77.0$

Show answer
Correct Option: B
Explanation:

Diameter =14

Radius =7
Area=$\pi R^2$
    $=\frac{22}{7}.7,7$
    $=22\times7 sq. inch$
Area of one piece=$\dfrac{1}{8}.22\times 7$
                $=\dfrac{11}{4}\times 7$
               $=19.25 sq.inch$

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. $2.62$

Show answer
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

Show answer
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. $60^\circ$

Show answer
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 $=\cfrac{7\pi}{6}\times \cfrac{180}{\pi}= 210^{\circ}$

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