Tag: area of sectors and segments

Questions Related to area of sectors and segments

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. $r=\sqrt{\dfrac{360A}{S\pi}}$

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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. $14.4\ m^{2}$

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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 $cm^2$

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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

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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

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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

The length of a minute hand of a wall clock is $8.4\ cm$. Find the area swept by it in half an hour.

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $100\ cm^{2}$

  2. $110.88\ cm^{2}$

  3. $120\ cm^{2}$

  4. $130\ cm^{2}$

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Correct Option: B
Explanation:

We know that minute hand covers $180^{o}$ in half an hour, which is a semicircle, hence area is

 $\Rightarrow \dfrac{1}{2}(\pi)(r^{2})=0.5\times3.1428\times(8.4)^{2}=110.88 \,cm^{2}$

The area of a sector of angle p (in degrees) of a circle with radius R is

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $\displaystyle \frac{p}{360} \times 2 \pi R$

  2. $\displaystyle \frac{p}{180}\times \pi R^2$

  3. $\displaystyle \frac{p}{720} \times 2 \pi R$

  4. $\displaystyle \frac{p}{720} \times 2 \pi R^2$

Show answer
Correct Option: D
Explanation:

Area of a sector with angle $p = \dfrac{p}{360} \times \pi \times R^2$ ,which matches with option D.

Find the area of sector whose length is $30\ \pi$ cm and angles of the sector is $40^o$.

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $2125\ \pi $ sq. cm

  2. $2225\ \pi $ sq. cm

  3. $2025\ \pi $ sq. cm

  4. $2200\ \pi $ sq. cm

Show answer
Correct Option: C
Explanation:
As we know that,
$1° = \cfrac{\pi}{180}$

$\therefore 40° = \cfrac{\pi}{180} \times 40 = \cfrac{2 \pi}{9}$

Let $S$ be the length of the arc and $A$ be the area of the corresponding sector.

Given that length of arc $\left( S \right) = 30 \pi \; cm$

As we know,
$S = r \theta$

$\Rightarrow 30 \pi = r \left( \cfrac{ \pi}{9} \right)$

$\Rightarrow r = 135 \; cm$

$\therefore$ Area of corresponding seector $\left( A \right) = \cfrac{1}{2} {r}^{2} \theta$

$\Rightarrow A = \cfrac{1}{2} {\left( 135 \right)}^{2} \left( \cfrac{2 \pi}{9} \right) = 2025 \pi \; {cm}^{2}$

The crescent shaded in the diagram, is like that found on many flags. $PSR$ is an arc of a circle, centre $O$ and radius $24.0$ cm. Angle POR $=$ $48.2^{\circ}$.
$PQR$ is a semicircle on $PR$ as diameter, where $PR$ $=$ $19.6$ cm
$[\pi = 3.14] [\cos 24.1 = 0.91]$

The area of the crescent is

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. $116.4$ cm$^2$

  2. $123.4$ cm$^2$

  3. $112.2$ cm$^2$

  4. $23.4$ cm$^2$

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Correct Option: B
Explanation:

Length PQR $= \displaystyle  \frac{1}{2} \times 2 \pi r = \frac{1}{2} \times 2 \times 3.14 \times 9.8 = 30.772 cm$
Length PSR $ = \displaystyle \frac{\theta}{360} \times 2 \pi = \frac{48.2}{360}\times 2 \pi r= \frac{48.2 }{360} \times 2 \times 3.14 \times24$
$= 20.1797 cm$
Perimeter of crescent $= 30.772 + 20. 1797 = 51$
Area of sector POR $= \displaystyle \frac{\theta}{360} \times \pi \times r^2 = \frac{48.2}{360} \times 3.14 \times 14^2$
$= 242.16 cm^2 = 242 cm^2$
In triangle POR height, $ h = 24 cos24.1 = 21.91 cm$
Area $= \displaystyle \frac{1}{2} bh = \frac{1}{2} \times 19.6 \times 21.91= 214.70 cm^2$
$\therefore $ Area of shape PSR remaining $= 242.16 - 214.70 = 27.46 cm^2$
Area of semicircle $= \displaystyle \frac{1}{2} \times \pi \times 9.8^2 = 150.859$
$\therefore$ Area of crescent $=150.859 - 27.46 = 123.4 cm^2$

If the area of a sector of a circle is $\dfrac{5}{18}$th of the area of that circle, then the central angle of the sector is 100. Is it true or false?

maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

  1. True

  2. False

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Correct Option: A
Explanation:

For $180^\circ$, we have $\cfrac{1}{2}$ of the total area.

Hence, for $\cfrac{5}{18}^{th}$ of the total area, we have $360 \times \cfrac{5}{18} = 100^\circ$