Tag: sector of a circle

Questions Related to sector of a circle

Points $A,B,C $ are on a circle, such that $m(arc AB)=m(arc BC)=^o$. No point, except point $B$, is common to the arcs.which is the type of $\triangle ABC$?

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. Equilateral triangle

  2. Scalene triangle

  3. Right angled triangle

  4. Isosceles triangle

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Correct Option: A

Find the area of a sector of a circle with radius $6$cm if angle of the sector is $60^o$.

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. $6\pi$

  2. $7\pi$

  3. $3\pi$

  4. $5\pi$

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Correct Option: A
Explanation:
Area of sector of circle $=\dfrac{\theta}{360}\times \pi r^2$
$=\dfrac{60}{360}\times \pi\times (6)^2$
$=\dfrac{1}{6}\times \pi\times (36)$
$=6\pi$.

Consider a circle with unit radius. There are seven adjacent sectors, $S _1, S _2, S _3, ............ S _7$, in the circle such that their total area is $\dfrac {1}{8}$ of the area of the circle. Further, the area of the $j^{th}$ sector is twice that of the $(j-1)^{th}$ sector, for $j$ $=$ $2, ........... 7$. What is the area of sector $S _1?$

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. $\displaystyle \frac{\pi }{508}$

  2. $\displaystyle \frac{\pi }{2040}$

  3. $\displaystyle \frac{\pi }{1016}$

  4. $\displaystyle \frac{\pi }{1524}$

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Correct Option: C
Explanation:

Let the area of thesector S$ _1$ be x units. Then, the area of the corresponding sectors shall be 2x, 4x, 8x, 16x, 32x and 64x. The total area then shall be 127x units. This is $\displaystyle \frac{1}{8}$ of the total area of the circle. 

Hence, the total area of the circle will be $127x \times 8 = 1,016 x\ units.$
$\Rightarrow 1016 x = \pi (1)^2 \Rightarrow x = \pi/1016$
Hence area of sector $S _1 $ is $\pi / 1016$

The angle subtended by the chord AB in the minor arc of S is - 

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. $\dfrac{3 \pi}{4}$

  2. $\dfrac{5 \pi}{6}$

  3. $\dfrac{2 \pi}{3}$

  4. $\dfrac{ \pi}{4}$

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Correct Option: A

The length of minor arc $\overset{\frown}{AB}$ of a circle is $\dfrac{1}{4}$ of its circumference, then the measure of the angle subtended by the minor arc $\overset{\frown}{AB}$ will be ....

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. 30

  2. 45

  3. 90

  4. 60

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Correct Option: A

Let a semicircle with centre O and diameter AB. Let P and Q be points on the semicircle and R be a point on AB extended such that OA =QR < PR. If $\widehat{POA} = 102^0$ then $\widehat{PRA} $ is 

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. $51^0$

  2. $34^0$

  3. $25.5^0$

  4. not possible to be determined

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Correct Option: D
Explanation:


Since R can be any point between A & R' & hence its corresponding point Q will lie on the arc AQ'. Hence, $\angle$ PRA can not be uniquely determined.

With a given centre and a given radius,only one circle can be drawn.

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. True

  2. False

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Correct Option: A

If angle of sector is $x^o$, then formula used to calculate area is

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. $\dfrac{x^o}{360}\times \pi r^2$

  2. $2\dfrac{x^o}{360}\times \pi r$

  3. $\dfrac{x^o}{180}\times \pi r^2$

  4. $2\dfrac{x^o}{360}\times r^2$

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Correct Option: A
Explanation:

If angle of sector is $x^o$ then formula used to calculate is $\dfrac{x^o}{360}\times \pi r^2$.

If the circumference of a circle is $8$ units and arc length of major sector is $5$ units then find the length of minor sector.

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. $3$ units

  2. $5$ units

  3. $7$ units

  4. None of these

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Correct Option: A
Explanation:

Length of major arc + Length of minor arc = Circumference

Length of Minor arc $= 8 – 5 = 3$ units 

The angle subtended at the centre of a circle of radius $3cm$ by an arc of length $1cm$ is:

maths circle and its elements angle subtended by arc sector of a circle arcs and sectors

  1. $\cfrac { { 30 }^{ o } }{ \pi } $

  2. $\cfrac { { 60 }^{ o } }{ \pi } $

  3. ${ 60 }^{ o }$

  4. None of the above

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Correct Option: B
Explanation:

Angle subtended at the centre  of circle is $\theta =\dfrac { l }{ r }$ 

$\Rightarrow \theta =\dfrac { 1 }{ 3 }$  
Now, $\pi$ radian $ =180^{o}$ 
$\Rightarrow \frac { 1 }{ 3 }$ radian $=180^{o}\times \dfrac { 1 }{ 3\pi  } =\dfrac { 60^{o} }{ \pi  }$ 
Hence, option B is correct.