Mathematics · Quantitative Aptitude

Circle and Arc Properties

117 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 mathematics and statistics angle and their measurement degree measure of angle measure of angle radians or degrees

The value of $\displaystyle 144^{\circ}$ in circular measure is ___ 

  1. $\displaystyle \frac{3\pi ^{c}}{4}$
  2. $\displaystyle \frac{2\pi ^{c}}{3}$
  3. $\displaystyle \frac{4\pi ^{c}}{5}$
  4. $\displaystyle \frac{5\pi ^{c}}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$ {144}^{0} = {144}^{0} \times \dfrac {{\pi}^{c}}{{180}^{0}} = \dfrac {4{\pi}^{c}}{5} $

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

The greatest possible number of points of intersection of 8 straight lines and $4$ circles is $104$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

According to question,

There are 8 lines, for two lines meet in a point,

=>  $^8C _2\times 1= \dfrac{8.7}{1.2}=28$

Line and circle meet in two points,

=>$(^8C _1\times ^4C _1) \times 2 =64$

Two circles meet in two points,

=>  $(^4C _2)\times2 =\dfrac{4.3}{1.2}.2= 104$

Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

There are 6 equally spaced points A, B, C, D, E and F marked on a circle with radius R. How many convex pentagons of distinctly different areas can be drawn using these points advertises?

  1. $^6P _5$
  2. $1$
  3. $55$
  4. $42$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For any 5 points chosen from 6 equally spaced points on a circle, the area of the resulting convex pentagon is determined by the relative positions of the points. Due to the symmetry of the points on the circle, all convex pentagons formed by choosing 5 out of 6 points are congruent and thus have the same area.

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

The region between an arc and two radii joining the centre to the end points of the arc is called

  1. sector

  2. segment

  3. semicircle

  4. non of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The region between an arc and two radii joining the center to the end points of the arc is called sector.The minor are corresponds to minor sector  and  major arc correspond to major sector.

Multiple choice maths circle and its elements angle subtended by arc sector of a circle arcs and sectors
$\triangle ABC$ is inscribed in a circle. Point $P$ lies between $A$ and $C$, whereas point $Q$ lies between $B$ and $C$. If $m(\text{arc}\, APC) = 60^\circ$ and $\angle BAC = 80^\circ$, find $m(\text{arc}\, BQC)$.
  1. $180^\circ$
  2. $90^\circ$
  3. $160^\circ$
  4. $120^\circ$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

By inscribed angle theorem, 

$ \cfrac 12 m\angle BAC = m(arc BQC)$
$m(arc BQC) = 2 \times \angle BAC$
$\therefore m(arc BQC) = 2 \times 80^o = 160^o$

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.