Tag: circle measures

Questions Related to circle measures

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

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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

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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 $360 \times \cfrac{5}{18} = 100^\circ$

An arc AB of a circle subtends an angle x radians at the centre O of the circle. Given that the area of the sector AOB is equal to the square of the length of the arc AB, then the value of x?

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

  1. $\dfrac{1}{3}$

  2. $\dfrac{1}{4}$

  3. $\dfrac{1}{5}$

  4. $\dfrac{1}{2}$

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Correct Option: D

A wire of length $20\ cm$ can be bent $n$ the form of a sector then its maximum area is 

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

  1. $15\ sq.cm$

  2. $25\ sq.cm$

  3. $5\ sq.cm$

  4. $none$

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Correct Option: A

ABC is a right angel triangle right angled at vertex A. A circle is drawn to touch sides AB and AC at points P and Q respectively such that other end points of diameters passing through P and Q lie on side BC. If AB = 6. then the area of circular sector which lies outside the triangle is :

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

  1. $\pi -2$

  2. $\pi -3$

  3. 4

  4. $\pi +2$

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Correct Option: A

The area of the sector of circle ${x}^{2}+{y}^{2}=16$ and the line $y=x$ in the first quadrant is 

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

  1. $8\pi sq.units$

  2. $\pi sq.units$

  3. $4\pi sq.units$

  4. $2\pi sq.units$

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Correct Option: A

The area of a sector whose perimeter is four times its radius (r units)is

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

  1. $\sqrt{r}\,sq.\,units$

  2. ${r}^{4}\,sq.\,units$

  3. ${r}^{2}\,sq.\,units$

  4. $\displaystyle \frac {{r}^{2}}{r}\,sq.\,units$

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Correct Option: A