Mathematics · Quantitative Aptitude

Circle Geometry

250 Questions

Circle geometry focuses on the properties and dimensions of circles, including radius, diameter, circumference, and area calculations. Questions often involve concentric circles, sectors, and inscribed shapes. This topic frequently appears in quantitative aptitude sections of major competitive exams.

Area and circumferenceConcentric circlesCircle sectorsInscribed polygonsSemicircle area

Circle Geometry Questions

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

A circular disc of radius $10 cm$ is divided into sectors with angles $120^o$ and $150^o$, then the ratio of the area of two sectors is

  1. $4 : 5$
  2. $5 : 4$
  3. $2 : 1$
  4. $8 : 7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of sector formed from angle $\theta=\dfrac{\theta}{360^\circ}\pi r^2$, where $r$ is the radius of the circle
Now, if angle is $120^\circ$, $150^\circ$ then the ratio of area of sector will be

$\Rightarrow\dfrac{\dfrac{120^\circ}{360^\circ}\pi r^2}{\dfrac{150^\circ}{360^\circ}\pi r^2}$

$\Rightarrow \dfrac{4}{5}$
Hence, the required ratio is $4:5$. 

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

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 $S _1?$

  1. $\displaystyle \frac{\pi }{508}$
  2. $\displaystyle \frac{\pi }{2040}$
  3. $\displaystyle \frac{\pi }{1016}$
  4. $\displaystyle \frac{\pi }{1524}$
Reveal answer Fill a bubble to check yourself
C Correct answer
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 $\pi / 1016$

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

A sector of a circle with sectorial angle of $\displaystyle 36^{\circ} $ has an area of 15.4 sq cm The length of the arc of the sector is

  1. $8.8 m$
  2. $4.4 m$
  3. $0.22 m$
  4. $0.44 m$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle \frac{36}{360}\times \frac{22}{7}r^{2}=15.4\Rightarrow r^{2} =\frac{15.4\times 5\times 7}{11}=49$
$\displaystyle \Rightarrow r=7$
$\displaystyle C=2\pi r=2\times \frac{22}{7}\times 7=44cm$
$\displaystyle =0.44 m$

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle
The diameter of a circle is $10$ cm, then find the length of the arc, when the corresponding central angle is $180^{\circ}$.  $(\pi =3.14)$
  1. $15.7$
  2. $16$
  3. $3.14$
  4. $18$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Radius of the circle $ = \dfrac {\text{Diameter}}{2} = 5 $ cm 


Length of an arc subtending an angle $ \theta  = \dfrac { \theta  }{ 360 }

\times 2\pi R $, where $R$ is the radius of the circle. 

So, length of the arc $ = \dfrac {180}{360} \times 2 \times 3.14\times 5  = 15.7 $ cm

Multiple choice maths circle measures length of an arc area of a sector of a circle sector and arc of a circle
The diameter of a circle is $10$ cm, then find the length of the arc, when the corresponding central angle is $144^{\circ}$.$(\pi =3.14)$
  1. $44$ cm
  2. $12.56$ cm
  3. $12$ cm
  4. $88$ cm
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Radius of the circle $ = \dfrac {Diameter}{2} = 5  cm $

Length of an arc subtending an angle $ \theta  = \dfrac { \theta  }{ 360 }

\times 2\pi R $ where R is the radius of the circle. 



So, length of the arc $ = \dfrac {144}{360} \times 2 \times 3.14 \times 5  = 12.56  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 from a circle of radius $21$ cm. The angle of the sector is $150^o$. Find the length of its arc and area.

  1. $27$ cm and $412.7cm^2$
  2. $36$ cm and $436.9cm^2$
  3. $45$ cm and $517.5cm^2$
  4. $55$ cm and $577.5cm^2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The length of arc $l$ and area $A$ of a sector of angle $\theta$ in a circle of radius $r$ are given by,


$l=\displaystyle\frac{\theta}{360^o}\times 2\pi r$

and $A=\displaystyle\frac{\theta}{360^o}\times \pi r^2$ respectively.

Here, $r=21$ cm and $\theta=150^0$


$\therefore l = \dfrac{150}{360}\times2\times\dfrac{22}7\times21 = 55$ cm

and 

$A = \dfrac{150}{36}\times\dfrac{22}7\times21^2 = \dfrac{1155}2 = 577.5\ {cm}^2$

Multiple choice maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents

Construct a regular pentagon inside a circle of radius $6\ cm$. The length of each side of the pentagon is: (approx.)

  1. $6\ cm$
  2. $7\ cm$
  3. $8\ cm$
  4. $9\ cm$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Each side of the pentagon makes an angle x at the center

$\implies 5x= 360 $

$x = 72$

Now lets consider side AB which is a chord to the circle

Let OP be a perpendicular to AB

$\implies AP = BP \implies AB = 2AP$

IN $\triangle OAP$

$\angle OPA = 90$

$\angle POA = \dfrac{x}{2} = \dfrac{72}{2} = 36$

$\sin 36 = \dfrac{AP}{OA}$

$AP = 0.6 \times 6 = 3.6$

$AB = 2 \times 3.6 = 7cm$

Multiple choice maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

Relation between circumradius and number of sides is given by-

  1. $Area=\dfrac{r^2n\sin(\dfrac{360}{n})}{3}$
  2. $Area=\dfrac{r^2n\sin(\dfrac{360}{n})}{2}$
  3. $Area=\dfrac{r^2n\cos(\dfrac{360}{n})}{2}$

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

The area of a regular polygon with n sides and circumradius r is given by n * (1/2 * r^2 * sin(360/n)). This formula is derived by summing the areas of n isosceles triangles with sides r and included angle 360/n.

Multiple choice maths percentages finding one number as percentage of another money and metric measures as percentage expressing one quantity as a percentage of another

When the circumference of a circle decreases from $3\, \pi$ to $\pi$ , its area decreases by

  1. $16\, \displaystyle \frac{2}{3}$ %
  2. $66\, \displaystyle \frac{2}{3}$ %
  3. $88\, \displaystyle \frac{8}{9}$ %
  4. $12\, \displaystyle \frac{1}{2}$ %
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Ratio of circumference = 3 : 1
Ratio of radii = 3 : 1
$\therefore$ ratio of areas $=\, 3^2\, : 1^2\, 9\, :\, 1$
% decrease in area $=\, \displaystyle \frac{8}{9}\, \times\, 100$
$=\, 88\, \displaystyle \frac{8}{9}$ %