Tag: circle measures

Questions Related to circle measures

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. $528$

The areas of two concentric circles forming a ring are 154 sq cm and 616 sq cm The breadth of the ring is

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

  1. $21 cm$

  2. $56 cm$

  3. $14 cm$

  4. $7 cm$

Show answer
Correct Option: D
Explanation:

Breadth of the ring is equal to the difference between the radius of the outer circle and the radius of the inner circle
Given the area of outer circle=616$\displaystyle cm^{2}$
$\displaystyle \Rightarrow  \pi r _{2}^{2}=616 cm^{2}$
$\displaystyle \Rightarrow r _{1}^{2}=\frac{616\times 7}{22}=196$
$\displaystyle \therefore r _{1}=14 cm $
and the area of the inner circle $\displaystyle =154 cm^{2}$
$\displaystyle \Rightarrow \pi r _{2}^{2}= 154$
$\displaystyle \Rightarrow r _{2}^{2}=\frac{154\times 7}{22}=49$
$\displaystyle \therefore r _{2}=7 cm.$
$\displaystyle \therefore $ The required answer $\displaystyle =r _{1}-r _{2}=14-7=7 cm.$

If the difference between the circumference and radius of a circle is 37 cm, then the area of the circle is

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

  1. $111 cm^2$

  2. $148 cm^2$

  3. $259 cm^2$

  4. $154 cm^2$

Show answer
Correct Option: D
Explanation:

Consider the difference between circumference and radius.


Taking$\pi $ as $3.14$ will give an approximate answer. we can do it by taking $22/7$ as wel


Let, the radius be$ r cm$


Circumference will be,

$2r=2\times 3.14r=6.28r$


ATQ,  $6.28r-1.r=37$

$5.28r=37$

 $r=\dfrac{37}{5.28}$

$r=$approx. $7 cm$ 

Area$=\pi {{r}^{2}}=3.14\times 7\times 7=154c{{m}^{2}}$


Hence, this is the answer.

If the circumference of a circle is reduced by 50%, then the area will be reduced by

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

  1. 50%

  2. 25%

  3. 75%

  4. 12.5%

Show answer
Correct Option: C
Explanation:
Let the original radius be $r$.
So, the area of circle $=\pi r^2$             $....... (1)$

Since, the circumference of the circle is reduced by $50\%$.
It means that the radius of the circle is also reduced by $50\%$.

Then,
The new radius $=0.5r$

Therefore, the new area
$=\pi (0.5r)^2$
$=0.25\pi r^2$

Therefore, the required $\%$
$=\dfrac{\pi r^2-0.25\pi r^2}{\pi r^2}\times 100$
$=\dfrac{0.75\pi r^2}{\pi r^2}\times 100$
$=75\%$

Hence, this is the answer.

The diameter of semi circular field is $49$ m. What the cost of fencing the plot of Rs. $10$ per metre?

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

  1. Rs. $ 1259.3$

  2. Rs. $1260$

  3. Rs. $1250$

  4. None

Show answer
Correct Option: A
Explanation:

The perimeter of a semi circle is $= \pi r+2r$

Here $r:$ radius of the semi circle
Given that $2r=49$
$\Rightarrow r= \dfrac { 49 }{ 2 } $
Perimeter of the semi circle $ = (\pi +2)r= (\pi +2)\dfrac { 49 }{ 2 } = 125.93$ m 
Cost of fencing the plot per metre $=$ Rs. $10$
Total cost of fencing $= 125.93\times10=$ Rs. $1259.3$.

Find the area of the semicircle whose radius is $4$ cm.

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

  1. $25.12$ sq. cm

  2. $15.12$ sq. cm

  3. $12.12$ sq.cm

  4. $21.12$ sq. cm

Show answer
Correct Option: A
Explanation:

Given, radius $=4$ cm
We need area of semicircle with the given radius
Area of a semicircle $=$ $\dfrac{1}{2}\pi  r^2$
$=$ $\dfrac{1}{2}\pi\times  (4)^2$
$= 25.12$ sq. cm

Find the circumference of the semicircle whose diameter is $4\ cm.$

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

  1. $3.46\ cm$

  2. $5.89\ cm$

  3. $6.28\ cm$

  4. $10.28\ cm$

Show answer
Correct Option: D
Explanation:

Diameter $= 2 \times$ radius
radius $= \dfrac{4}{2}=2$ $cm$
Circumference of the semicircle $= \pi r+ 2r$
= $3.14 \times  2 +4= 10.28$ $cm$

A window in the shape of a semicircle has a radius of $20\ cm$. Find the area of the semicircle.

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

  1. $128\ cm$

  2. $428\ cm$

  3. $628\ cm$

  4. $290\ cm$

Show answer
Correct Option: C
Explanation:

Area of a semicircle $=\dfrac{1}{2}\pi r^2$
$=\dfrac{1}{2}\pi (20)^2$
$=628\ cm^2$