Tag: semicircle and ring

Questions Related to semicircle and ring

The area of the concentric circles are $962:5\ cm^{2}$ and $1368\ cm^{2}$. Find the width of the ring formed by them. 

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. $2:1\ cm$

  2. $1 cm$

  3. $5:5\ cm$

  4. $3:5\ cm$

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Correct Option: B

Find the area of a ring shaped region enclosed between two concentric circles of radii $20$ cm and $15$ cm.

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. 550 $cm^{2}$

  2. 425 $cm^{2}$

  3. 496 $cm^{2}$

  4. 810 $cm^{2}$

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Correct Option: A
Explanation:

The area of a ring shaped region$=\pi(20)^2-\pi(15)^2$
                                                    $=(400-225)\times \dfrac{22}{7}$
                                                    $=(175)\times \dfrac{22}{7}$
                                                   $ =550$ sq. cm

A circular park has a path of uniform width around it. The difference between outer and inner circumference of the circular path is $132\ m$. Its width is _____ $\displaystyle \left(\pi=\frac{22}{7}\right)$

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. $22\ m$

  2. $20\ m$

  3. $21\ m$

  4. $24\ m$

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Correct Option: C
Explanation:

Given $\displaystyle 2\pi r _{2}-2\pi r _{1}=132$
$\displaystyle \Rightarrow r _{2}-r _{1}=\frac{132}{2\pi }=\frac{132\times 7}{44}=21\ m$

The inner circumference of a circular track $14\ m$ wide is $440\ m$. The radius of the outer circle is

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. $70\ m$

  2. $56\ m$

  3. $77\ m$

  4. $84\ m$

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Correct Option: D
Explanation:

Inner circumference $= \displaystyle 2\pi r=2\times \frac{22}{7}\times r=\frac{44r}{7}$
Given $\displaystyle \frac{44r}{7}=440\Rightarrow r=\frac{440\times 7}{44}=70\ m$
$\displaystyle \therefore $ Radius of outer circle $= 70\ m + 14\ m = 84\ m$ 

If the perimeter of a semi-circle is $36\ cm$. What will be its diameter?

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. $88\ cm$

  2. $22\ cm$

  3. $28\ cm$

  4. $14\ cm$

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Correct Option: D
Explanation:
Perimeter of semicircle$=36cm$
$\pi r+d=36$
$\Rightarrow \pi r+2r=36$
$\Rightarrow r\left( \cfrac { 22 }{ 7 } +2 \right) =36$
$\Rightarrow \cfrac { 36 }{ 7 } \times r=36$
$\Rightarrow r=7$
$\therefore $ Diameter $2r=2\times 7=14cm$

If a wire is bent into the shape of a square the area of the square is 81 sq cm .When the wire is bent into a semi circular shape; what is the area of the semicircle? $\displaystyle \left ( \pi =\frac{22}{7} \right )$

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. $\displaystyle 77:\text{cm}^{2}$

  2. $\displaystyle 73:\text{cm}^{2}$

  3. $\displaystyle 37:\text{cm}^{2}$

  4. $\displaystyle 33\ \text{cm}^{2}$

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Correct Option: A
Explanation:

Let '$a$' be the length of each side of the square
Then $\displaystyle a^{2}=81\Rightarrow a=9$ cm
Length of wire $=$ Perimeter of square
=$ 4a= 36$ cm
$\displaystyle \Rightarrow $ Circuference of semicircle $= 36$ cm
$\displaystyle \Rightarrow \pi r+2r=36$

$ \displaystyle \Rightarrow r\left ( \pi +2 \right )=36$
$\displaystyle \Rightarrow  r =\dfrac{36}{\pi +2}=\dfrac{36}{\dfrac{22}{7}+2}=\dfrac{36\times 7}{\left ( 22+14 \right )}=\dfrac{36\times 7}{36}$ cm $=7$ cm
$\displaystyle \therefore \ \text{Area of the semicircle}=\frac{1}{2}\pi r^{2}$
$\displaystyle =\dfrac{1}{2}\times \dfrac{22}{7}\times 7\times \text{cm}^{2}=77\text{cm}^{2}$

If the radii of two concentric circles are $15\ cm$ and $13\ cm$ respectively then the area of the circulating ring in sq cm will be

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. $176$

  2. $178$

  3. $180$

  4. $200$

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Correct Option: A
Explanation:

$R = 15\ cm, r = 13\ cm$
Area of the circulating ring 
$\displaystyle = \pi \left(R^2-r^2\right)$

$\displaystyle =\pi \left ( R+r \right )\left ( R-r \right )$
$\displaystyle =\frac{22}{7}\left ( 15+13 \right )\times \left ( 15-13 \right )$
$\displaystyle \frac{22}{7}\times 28\times2$
$= 176$ sq cm

A semicircle is drawn with $AB$ as its diameter. From $C$ a point on $AB$ a line perpendicular to $AB$ is drawn meeting the circumference of the semicircle at $D$. Given that $AC = 2\ cm$ and $CD = 6\ cm$ the area of the semicircle is :

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. $\displaystyle 32\pi $

  2. $\displaystyle 50\pi $

  3. $\displaystyle 40\pi $

  4. $\displaystyle 36\pi $

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Correct Option: B
Explanation:

Let O be the centre of the circle. Then, $OA = OB = OD= r$
Now, $OC = r - 2$
and $CD = 6$
Thus, in $\triangle ODC$
$OC^2 + CD^2 = OD^2$
$(r - 2)^2 + 6^2 = r^2$
$r^2 + 4 - 4r + 36 = r^2$
$4r = 40$
$r = 10$ $cm$
Area of semicircle $= \dfrac{\pi r^2}{2} = \dfrac{\pi (10)^2}{2} = 50 \pi$

If a wire is bent into the shape of a square, then the area of the square is $81\ cm^{2}$. When the same wire is bent into a semi-circular shape, then the area of the semi circle will be

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. $22\ cm^{2}$

  2. $44\ cm^{2}$

  3. $77\ cm^{2}$

  4. $154\ cm^{2}$

Show answer
Correct Option: C
Explanation:

Let the side of the square be $a$ cm

Thus, perimeter is $4a$ and area of the square is $a^2$
$\therefore a^2=81$
$\Rightarrow a=\sqrt{81}$
$\Rightarrow a=9$
Thus, perimeter $=4 \times 9$ $=36$ cm
Now, wire is bent into a semicircle.
Therefore, $2r+πr=36$
$⇒r(π+2)=36$ $
$⇒r=\cfrac{36}{\dfrac{22}{7}+2}$

$⇒r=\cfrac { 36 }{\dfrac{36}{7}}$
$⇒r=7$ cm

Area of semicircle $=\cfrac{1}{2}πr^2$
$=\cfrac{1}{2}×\cfrac{22}{7}×7×7$
$=77\ cm^2$

The inner circumference of a circular tracks is $220\ m$. The track is $7$ $m$ wide everywhere. Calculate the cost of putting up a fence along the outer circle at the rate of Rs. $2$ per metre. Use $\pi=\displaystyle\frac{22}{7}$

maths circle measures area between two concentric circles the area of ring semicircle and ring

  1. Rs. $947$

  2. Rs. $726$

  3. Rs. $612$

  4. Rs. $528$

Show answer
Correct Option: D
Explanation:

Circumference of inner side = $220 m$


$\Rightarrow 2\pi r=220\Rightarrow r=\dfrac { 220\times 7 }{ 44 } =35m$


Now, width of track  $=7m$

$\therefore $ Outer radius $=35+7 = 42 m$

Therefore outer circumference  $=2\pi R=2\times \dfrac { 22 }{ 7 } \times 42=264m$

$\therefore $ Cost of fencing $=$ Rs. $\left( 264\times 2 \right) $ $=$ Rs. $528$