Mathematics · Quantitative Aptitude

Circle and Arc Properties

115 Questions

Circle and arc properties involve calculating arc lengths, understanding radius relationships, and solving geometric proofs. These geometry concepts are crucial for quantitative aptitude tests. Review these questions to improve your spatial reasoning and accuracy.

Arc length calculationsCircle theoremsRadius and diameterInscribed polygonsCentral angles

Circle and Arc Properties Questions

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

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 ....

  1. 30

  2. 45

  3. 90

  4. 60
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circumference of a circle corresponds to an angle of 360 degrees at the center. Since the minor arc length is 1/4 of the total circumference, the central angle subtended by this arc is (1/4) * 360 degrees = 90 degrees.

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

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 

  1. $51^0$
  2. $34^0$
  3. $25.5^0$

  4. not possible to be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
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.

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

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

  1. $\cfrac { { 30 }^{ o } }{ \pi } $
  2. $\cfrac { { 60 }^{ o } }{ \pi } $
  3. ${ 60 }^{ o }$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

In a circle of radius 21 cm an arc subtends an angle of $\displaystyle 56^{\circ} $ at the centre of the circle. The length of the arc is

  1. $20.53$ cm
  2. $17.53$ cm
  3. $15.53$ cm
  4. $16.53$ cm
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\displaystyle \theta =56^{\circ},r=21cm$
Length. of $\displaystyle AB=\dfrac{56^{\circ}}{360^{\circ}}\times 2\times \dfrac{22}{7}\times 21$
$\displaystyle =\dfrac{616}{30}=20.53cm$

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

What is the length of arc AB making angle of $126^0$ at center of radius $8$?

  1. $2.6\displaystyle \pi $
  2. $5.6\displaystyle \pi $
  3. $7.6\displaystyle \pi $
  4. $\displaystyle \frac{1}{2}\pi $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Setting a proportion 
AB : OB : : 126 : 360
$\displaystyle \frac{\overline{AB}}{2\pi r}=\frac{126}{360}$
$\displaystyle \overline{AB}=\left ( \frac{126}{360} \right )\times 2\pi r$
$\displaystyle \overline{AB}=\left ( \frac{126}{360} \right )\times 2\pi \times 8$
$\displaystyle \overline{AB}=5.6\pi $

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

If an arc of a circle of radius 14 cm subtends an angle of $60^{\circ}$ at the centre, then the length of the arc is $\displaystyle \frac{44}{3} cm$.

  1. True

  2. False

  3. Niether

  4. Either
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Arc length = (theta / 360) * 2 * pi * r. Here, (60 / 360) * 2 * (22 / 7) * 14 = (1 / 6) * 2 * 22 * 2 = 88 / 6 = 44 / 3 cm. The statement is true.

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

The length of an arc of a sector of a circle of radius r units and of centre angle $\theta$ is $\displaystyle \frac{\theta}{360^{\circ}} \times \pi r^2$.

  1. True

  2. False

  3. Neither

  4. Either
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The formula provided, (theta / 360) * pi * r^2, is the formula for the area of a sector, not the length of an arc. The arc length formula is (theta / 360) * 2 * pi * r.

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

Length of an arc of a circle with radius $r$ and central angle $\theta$ is(angle in radians):

  1. $\dfrac{r\times \theta}{360^{o}}$
  2. $\dfrac{r\times \theta}{180^{o}}$
  3. $\dfrac{r\times \theta}{90^{o}}$
  4. $r\times \theta$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let $r$ be the radius of a circle and $\theta$ be the central angle
Length of an arc of the sector $=r\times \theta$
Hence, length of an arc of a circle $=r\times \theta$.
Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

The perimeter and area of a sector are $18\;cm$ and $20\;sq.\,cm$ respectively. Then the length of the arc is:

  1. $10\;cm\;or\;8\;cm$
  2. $10\;cm\;or\;5\;cm$
  3. $10\;cm\;or\;4\;cm$
  4. $20\;cm\;or\;2\;cm$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$l+2r=18$
$\displaystyle\frac{lr}{2}=20$
$lr=40$
$l=\displaystyle\frac{40}{r}$
$18=\displaystyle\frac{40}{r}+2r$
$r^2-9r+20=0$
$(r-4)\;\;(r-5)=0$
$r=4$ or $r=5$
$l=8$ or $l=10\;cm$

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle

A sector is cut off from a circle of radius $21$ cm The angle of the sector is $\displaystyle 120^{\circ} $ The length of its arc is [Take $\displaystyle \pi =\frac{22}{7} $]

  1. $40 cm$
  2. $44 cm$
  3. $35 cm$
  4. $28 cm$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given radius$(r)=21cm$ and the angle$(\theta)=120^\circ$

length of arc$=r\times \theta$

here $r=21cm,$ $\theta=120^\circ=\dfrac{120}{360}\times 2\pi$

length of arc = $\dfrac { \theta  }{ { 360 }^{ 0 } } \times 2\pi r=\dfrac { { 120 }^{ 0 } }{ { 360 }^{ 0 } } \times 2\times \dfrac { 22 }{ 7 } \times 21=44cm$