Tag: circle measures

Questions Related to circle measures

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} $

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