Tag: maths

Questions Related to maths

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. $72 m$

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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. $28 cm$

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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.