Mathematics · Quantitative Aptitude

Circle Geometry

232 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 measures area of a sector of a circle sector and arc of a circle area of sectors and segments

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 $(j - i)^{th}$ sector, for $j = 2, .... 7$. Find the area of the sector $S _{1}$

  1. $\dfrac {\pi}{1016}$
  2. $\dfrac {\pi}{986}$
  3. $\dfrac {\pi}{116}$

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

The total area of the seven sectors is (1/8) * pi * r^2. With r=1, total area = pi/8. The areas form a geometric progression: a, 2a, 4a, 8a, 16a, 32a, 64a. Sum = a(2^7 - 1)/(2 - 1) = 127a. 127a = pi/8, so a = pi / (127 * 8) = pi / 1016.

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 a sector of a circle of radius $28$cm and central angle $45^0$.

  1. $616 cm^{2}$
  2. $308 cm^{2}$
  3. $508 cm^{2}$
  4. $154 cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Radius of sector $=28 cm$

Control angle $=45^{ o }$
Area of sector $=\cfrac { \theta  }{ 360° } \times \pi { r }^{ 2 }$
$=\cfrac { 45° }{ 360° } \times \cfrac { 22 }{ 7 } \times 28\times 28\ =308\quad { cm }^{ 2 }$

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

If a sector of a circle of diameter 21 cm subtends an angle of $120^{\circ}$ at the centre, then what is its area ? 

  1. $115.5 \ cm^2$.
  2. $84 \ cm^2$.
  3. $85.5 \ cm^2$.
  4. $78 \ cm^2$.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Area of sector = $\cfrac{120}{360} \times \pi \times (\cfrac{21}{2})^2$

Thus area = $\cfrac{1}{3} \times \cfrac{22}{7} \times \cfrac{441}{4} = 115.5 cm^2$

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 sector of a circle of diameter $14 cm$ subtends an angle of $30^{\circ}$ at the centre, then its area is

  1. $49 \pi$
  2. $\displaystyle \frac{49 \pi}{12}$
  3. $\displaystyle \frac{242}{3\pi}$
  4. $\displaystyle \frac{121}{3\pi}$
Reveal answer Fill a bubble to check yourself
B,D Correct answer
Explanation

Area of a sector $=\dfrac{\theta}{360^0} \times \pi r^2 =\dfrac{30}{360} \times \pi (7)^2 = \dfrac{49 \pi}{12}$


Also, 
$ \dfrac{121}{3\pi}=\dfrac{121 \times 7}{3 \times 22} = \dfrac{49 \times 22}{12 \times 7} = \dfrac{49 \pi}{12}$

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

A circular disc of radius 10 cm is divided into sectors with  angles $120^{\circ}$ and $150^{\circ}$ 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 $\theta$ = $\frac{\theta}{260} \pi r^2$, where r is the radius of the circle
Now, if angle is 120, 150 then the ratio of area of sector will be:
= $\frac{\frac{120}{360} \pi r^2}{\frac{150}{360} \pi r^2}$
= $\frac{120}{150}$ = 4:5

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

The area of a sector of a circle of angle $\displaystyle 60^{\circ}$ is $\displaystyle \frac{66}{7}cm^{2}$ then the area of the corresponding major sector is

  1. $\displaystyle 14cm^{2}$
  2. $\displaystyle \frac{55}{7}cm^{2}$
  3. $\displaystyle \frac{110}{7}cm^{2}$
  4. $\displaystyle \frac{330}{7}cm^2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of a sector of a circle of radius 'r' and angle  $ \theta = \frac {

\theta  }{ 360 } \pi {r}^{2}$
Given, $ \frac { 60 }{ 360 } \times \frac {22}{7} \times {r}^{2} = \frac {66}{7}  {cm}^{2} $
$ {r}^{2} = 18 $
Now, area of sector with angle {300}^{o} $ = \frac

{ 300 }{ 360 } \times \frac {22}{7} \times {r}^{2} = \frac

{ 300 }{ 360 } \times \frac {22}{7} \times 18 = \frac {330}{7}  {cm}^{2} $




Multiple choice maths the trapezium rule approximation errors and approximations the need for approximation

The circumference of a circle is measured as $28 cm$ with an error of $0.01 cm$. The percentage error in the area is

  1. $\dfrac {1}{14}$
  2. $0.01$
  3. $\dfrac {1}{7}$

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

Circumference $C=2\pi r$
$\Rightarrow\displaystyle r=\frac{14}{\pi}$
Also, $\displaystyle \frac{dC}{dr}=2\pi$
Area of circle $A=\pi r^{2} $
$\Rightarrow\displaystyle A=\frac{{14}^{2}}{\pi}$
Also, $\displaystyle \frac{dA}{dr}=2\pi r$
$\displaystyle \Rightarrow \frac{dA}{dC}=r=\frac{14}{pi}$
Approximate error in $A$ is $\displaystyle dA=( \frac{dA}{dC}) \Delta C$
                           $\displaystyle=\frac{14}{\pi}\frac{1}{100}$
                            $\displaystyle=\frac{1}{1400}$ of A
Percentage error in $A \ \displaystyle =\frac{1}{14}\%$

Multiple choice maths the trapezium rule approximation errors and approximations the need for approximation

The circumference of a circle is measured as $56$ cm with an error $0.02$ cm. The percentage error in its area is

  1. $\dfrac {1}{7}$
  2. $\dfrac {1}{28}$
  3. $\dfrac {1}{14}$
  4. $\dfrac {1}{56}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Circumference of circle $C=2\pi r=56cm$


$\Rightarrow \displaystyle r=\frac{28}{\pi}$

Also, $\displaystyle \frac{dC}{dr}=2\pi$

Area of circle $A=\pi r^{2}$

$\displaystyle \frac{dA}{dr}=2\pi r$

$\Rightarrow\displaystyle \frac{dA}{dC}=r =\frac{28}{\pi}$

Approximate error in A $=\displaystyle dA=(\frac{dA}{dC})\Delta C$

                                      $= r (0.02)$

$\Rightarrow\displaystyle \frac{dA}{A}= \frac{0.02}{\pi r}=\frac{1}{1400}$

Percentage error in A is $\displaystyle\frac{1}{14}$%

Multiple choice maths the trapezium rule approximation errors and approximations the need for approximation

If the error committed in measuring the radius of the circle is $0.05\%$, then the corresponding error in calculating the area is:

  1. $0.05\%$
  2. $0.025\%$
  3. $0.25\%$
  4. $0.1\%$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\dfrac { dr }{ r } =0.05\Rightarrow dr=(0.05)r$

Area of circle $=\pi r^2$
$\ A=\pi r^{ 2 }\Rightarrow \dfrac { dA }{ dr } =2\pi r\ dA=2\pi rdr\Rightarrow \dfrac { dA }{ A } =\dfrac { 2\pi rdr }{ \pi r^{ 2 } } =\dfrac { 2dr }{ r } \ \therefore \dfrac { dA }{ A } =2(0.05)^{ 2 }=0.1$
$\therefore$ Corresponding error in area $= 0.1\%$

Multiple choice maths tangents and intersecting chords other theorems related to circles touching circles angle made by a chord and a tangent

The equation to the locus of the point of intersection of any two perpendicular tangents to $x^{2}+ y^{2} = 4$ is

  1. $\mathrm{x}^{2}+\mathrm{y}^{2}=8$
  2. $\mathrm{x}^{2}+\mathrm{y}^{2}=12$
  3. $\mathrm{x}^{2}+\mathrm{y}^{2}=16$
  4. $\mathrm{x}^{2}+\mathrm{y}^{2}=4\sqrt{3}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The equation of the tangent to the circle $x^2+y^2=4$ is

$y=mx+2\sqrt{1+m^2}$
$P(h,k)$ lies on the tangent, then
$k-mh=2\sqrt{1+m^2}$
or, $(k-mh)^2=4(1+m^2)$
or, $m^2(h^2-4)-2mhk+k^2-4=0$
This is the quadratic equation in $m.$ Let $m _1$ and $m _2$ be roots
$m _1m _2=\cfrac{k^2-4}{h^2-4}=-1$
or, $k^2-4=-h^2+4$
or, $h^2+k^2=8$
Therefore, Equation to the locus of the intersection of any two perpendicular tangents is
$x^2+y^2=8$
Hence, A is the correct option.

Multiple choice

What is the name of the theorem that states that the area of a circle is equal to pi multiplied by the square of its radius?

  1. Brahmagupta's theorem

  2. Pythagorean theorem

  3. Euler's theorem

  4. Area of a Circle Formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Area of a Circle Formula provides a method for calculating the area of a circle using its radius.

Multiple choice

What is the name of the mathematical theorem that establishes the relationship between the circumference and diameter of a circle?

  1. The Pythagorean Theorem

  2. The Fundamental Theorem of Calculus

  3. The Intermediate Value Theorem

  4. The Circle Area Formula
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The Circle Area Formula provides a mathematical expression for calculating the area enclosed by a circle, involving the radius or diameter of the circle.

Multiple choice

What is the formula for the area of a circle?

  1. \(A = \pi r^2\)
  2. \(A = 2\pi r\)
  3. \(A = \pi d\)
  4. \(A = \frac{1}{2} \pi r^2\)
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The formula for the area of a circle is (A = \pi r^2), where (r) is the radius of the circle.

Multiple choice

What is the name of the theorem that states that the area of a circle is equal to pi times the square of its radius?

  1. Brahmagupta's Theorem

  2. Pythagorean Theorem

  3. Euler's Formula

  4. Descartes' Theorem
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Euler's Formula states that the area of a circle is equal to pi times the square of its radius.