Tag: area of a sector of a circle

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Questions Related to area of a sector of a circle

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

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

  1. $2125\ \pi $ sq. cm
  2. $2225\ \pi $ sq. cm
  3. $2025\ \pi $ sq. cm
  4. $2200\ \pi $ sq. cm
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $\left( A \right) = \cfrac{1}{2} {r}^{2} \theta$

$\Rightarrow A = \cfrac{1}{2} {\left( 135 \right)}^{2} \left( \cfrac{2 \pi}{9} \right) = 2025 \pi \; {cm}^{2}$
Multiple choice maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments

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

  1. $116.4$ cm$^2$
  2. $123.4$ cm$^2$
  3. $112.2$ cm$^2$
  4. $23.4$ cm$^2$
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $=150.859 - 27.46 = 123.4 cm^2$

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

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?

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $360 \times \cfrac{5}{18} = 100^\circ$

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

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?

  1. $\dfrac{1}{3}$
  2. $\dfrac{1}{4}$
  3. $\dfrac{1}{5}$
  4. $\dfrac{1}{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of sector = (1/2) * r^2 * x. Arc length = r * x. Given: (1/2) * r^2 * x = (r * x)^2. This simplifies to (1/2) * r^2 * x = r^2 * x^2. Dividing by r^2 * x (assuming x is not 0), we get 1/2 = x.

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

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 :

  1. $\pi -2$
  2. $\pi -3$

  3. 4
  4. $\pi +2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a geometry problem involving a circle inscribed in a right triangle. The sector area outside the triangle is calculated based on the geometry of the circle touching the sides. The result is pi - 2.

Multiple choice maths circle measures area of a sector of a circle sector and arc of a circle area of sectors and segments
The radius of a circle is $7 cm$, then area of the sector of this circle if the corresponding angle is $30^{\circ}$ is 
  1. $12.83 \,cm^2$
  2. $11.83 \,cm^2$
  3. $12.25 \,cm^2$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of a sector of a circle of radius '$r$' and angle $ = \dfrac { \theta  }{ 360 } \pi {r}^{2}$
Hence, area of the sector of the circle of  radius $ 7 $ cm and angle $ = \dfrac { 30 }{ 360 } \times \dfrac { 22 }{ 7 } \times 7 \times 7 = 12.83 \ \text{cm}^{2} $

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

The radius of a circle is $7 cm$, then area of the sector of this circle if the corresponding angle is:$210^{\circ}$ is 

  1. $88.83 \,cm^2$
  2. $87.83 \,cm^2$
  3. $89.83 \,cm^2$
  4. $86.83 \,cm^2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area of a sector of a circle of radius 'r' and angle  $ \theta = \dfrac { \theta  }{ 360 } \pi {r}^{2}$

Hence,area of the sector of the circle of  radius $ 7 $ cm and angle $ { 210 }^{

0 } = \dfrac { 210 }{ 360 } \times \dfrac { 22 }{ 7 } \times 7 \times

7\quad = 89.83  {cm}^{2} $