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 maths circle measures area between two concentric circles the area of ring semicircle and ring

A semicircle is drawn with $AB$ as its diameter. From $C$ a point on $AB$ a line perpendicular to $AB$ is drawn meeting the circumference of the semicircle at $D$. Given that $AC = 2\ cm$ and $CD = 6\ cm$ the area of the semicircle is :

  1. $\displaystyle 32\pi $
  2. $\displaystyle 50\pi $
  3. $\displaystyle 40\pi $
  4. $\displaystyle 36\pi $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let O be the centre of the circle. Then, $OA = OB = OD= r$
Now, $OC = r - 2$
and $CD = 6$
Thus, in $\triangle ODC$
$OC^2 + CD^2 = OD^2$
$(r - 2)^2 + 6^2 = r^2$
$r^2 + 4 - 4r + 36 = r^2$
$4r = 40$
$r = 10$ $cm$
Area of semicircle $= \dfrac{\pi r^2}{2} = \dfrac{\pi (10)^2}{2} = 50 \pi$

Multiple choice maths circle measures area between two concentric circles the area of ring semicircle and ring

Jackson measured the button on his shirt. Then he calculated that it has a semicircle of $25.12\ mm$. What is the button's radius? (Use $\pi = 3.14$).

  1. $1\ mm$
  2. $2\ mm$
  3. $3\ mm$
  4. $4\ mm$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of a semicircle $=\dfrac{1}{2}\pi r^2$
$ 25.12= \dfrac{1}{2}\pi r^2$
$ 50.24= \pi r^2  $

Using $\pi = 3.14$ (given)
$16 = r^2$
$r = 4\ mm$

Multiple choice maths circle measures area between two concentric circles the area of ring semicircle and ring

The area in ( ${cm^2}$) of the largest triangle that can be inscribed in a semicircle of radius r cm is 

  1. ${\cfrac{1}{3}\pi r^2}$
  2. ${2r^2}$
  3. ${r^2}$
  4. ${\cfrac{1}{2}\pi r^2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The largest triangle inscribed in a semicircle has a height = radius of the circle

base = diameter of the circle
$\therefore$ Area of a triangle $=\dfrac{1}{2}\times base\times height$
                                  $=\dfrac{1}{2}\times 2r \times r=r^2$
Hence, option C is correct.

Multiple choice maths circle measures area between two concentric circles the area of ring semicircle and ring

A wire in the shape of an equilateral triangle encloses an area $s$ sq. cm  If the same wire is bent to form circle, the area of the circle will be

  1. $\displaystyle \frac{\pi s^{2}}{9}$
  2. $\displaystyle \frac{3s^{2}}{\pi }$
  3. $\displaystyle \frac{3s}{\pi }$
  4. $\displaystyle \frac{3\sqrt{3}s}{\pi }$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of equilateral triangle $= s$ sq.cm
$\Rightarrow  \dfrac{\sqrt3}{4} a^2 = s$, [where $a$, the side of equilateral triangle]
$\Rightarrow a= \sqrt{\dfrac{4s}{\sqrt3}}$
Now perimeter of equilateral triangle $ 3\times a =3 \times\sqrt{\dfrac{4s}{\sqrt 3}}$ cm 
Circumference of circle $=$ perimeter of equilateral triangle
$\Rightarrow 2\pi r= 3 \times\sqrt{\dfrac{4s}{\sqrt 3}}$, [where $r$ the radius of circle]
Solve the above expression for $r$, we get 
$r= \dfrac{3}{2\pi} \times \sqrt{\dfrac{4s}{\sqrt 3}}$
Area of circle $=\pi r^2 = \pi \times \left ( \dfrac{3}{2\pi} \times \sqrt{\dfrac{4s}{\sqrt 3}} \right )^2$
After simplification, we get
Area of circle $=\dfrac{3s\sqrt3}{\pi}$ sq.cm

Multiple choice maths introduction to euclid's geometry conditional statements and converse euclid's postulates axioms, postulates and theorems euclid's fifth postulate

Identify the given statement: A circle can be described with any given center and radius.

  1. postulate

  2. conjectures

  3. theorem

  4. operation
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
A circle can be described with any given center and radius.
The given statement is postulate. A postulate is a statement that is accepted without proof. Axiom is another name for a postulate. 

For example, if you know that Pam is five feet tall and all her siblings are taller than her, you would believe her if she said that all of her siblings are at least five foot one. Pam just stated a postulate, and you just accepted it without grabbing a tape measure to verify the height of her siblings.
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 $ \displaystyle 120^{\circ}   $ and  $ \displaystyle 150^{\circ}   $ then  the ratio of the areas 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

Now $\frac{c}{circle}=\frac{120}{360}=\frac{1}{3}$

And $\frac{c}{circle}=\frac{150}{360}=\frac{5}{12}$
So sector with angle 120 and 150 is part $\frac{1}{3}$ and $\frac{5}{12}$
Now ratio of the area of two sectors =Ratio of central angle =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

Given, $\displaystyle A = \frac{S}{360}\times \pi r^2$
$A$ is the area of setor, $ S$ is the angle measure in degrees of the sector and $r$ is the radius of the circle. Find $r$ in terms of $A$ and $S$.

  1. $r=\dfrac{360A\pi}{S}$
  2. $r=\dfrac{360A}{S\pi}$
  3. $r=\sqrt{\dfrac{360A\pi}{S}}$
  4. $r=\sqrt{\dfrac{360A}{S\pi}}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To change the formula in terms of $A$ and $S$, Isolate $r^2$, we get the formula as
$\dfrac{A\times 360}{S\pi}=r^2$
Now taking square root on both the sides to get the value of $r$ in terms of $A$ and $S$.
$\therefore r=\sqrt{\dfrac{360A}{S\pi}}$

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 area of a sector of a circle is $\dfrac{5}{18}$th of the area of that circle, then the central angle of the sector is 100. Is it true or false?

  1. True

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

For $180^\circ$, we have $\cfrac{1}{2}$ of the total area.

Hence, for $\cfrac{5}{18}^{th}$ of the total area, we have $360 \times \cfrac{5}{18} = 100^\circ$

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

An arc AB of a circle subtends an angle x radians at the centre O of the circle. Given that the area of the sector AOB is equal to the square of the length of the arc AB, then the value of x?

  1. $\dfrac{1}{3}$
  2. $\dfrac{1}{4}$
  3. $\dfrac{1}{5}$
  4. $\dfrac{1}{2}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Area of sector = (1/2) * r^2 * x. Arc length = r * x. Given: (1/2) * r^2 * x = (r * x)^2. This simplifies to (1/2) * r^2 * x = r^2 * x^2. Dividing by r^2 * x (assuming x is not 0), we get 1/2 = x.

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

Arc of a sector is equal to-

  1. Length of arc $\times$ radius
  2. $ \displaystyle \frac{sector angle}{360^{\circ}}\times circumference of circle $
  3. $ \displaystyle \frac{sector angle}{360^{\circ}}\times (area of circle) $

  4. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The area of a sector is proportional to the angle it subtends at the center. It is (angle/360) * (area of the circle).

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

Points $A$ and $B$ lie on circle $O$ (not shown). $AO=3$ and $\angle AOB ={120}^{o}$. Find the area of minor sector $AOB$.

  1. $\dfrac{\pi}{3}$
  2. $\pi$
  3. $3 \pi$
  4. $9 \pi$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Area of a sector is given by $\cfrac{\theta}{360} \times \pi \times r^2$ where $\theta$ is the angle made by the sector, $r$ is the radius of the circle.

Here, $\theta = 120^o$ and $r = 3$
$\therefore$ area of minor sector $= \cfrac{120}{360} \times \pi \times 9 = 3\pi$

Multiple choice luminous intensity measurements physics

A lamp is hanging along the axis of a circular table of radius r. At what height should the lamp be placed above the table, so that the illuminance at the edge of the table is $\displaystyle \frac{1}{8}$ of that at its centre?

  1. r/2
  2. r/$\sqrt{2}$

  3. r/3
  4. r/$\sqrt{3}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$E _2 = \displaystyle \frac{1}{8} E _1$ or $\displaystyle \frac{1}{(r^2 + h^2)} \times \frac{h}{\sqrt{r^2 + h^2}} = \frac{1}{8} \frac{1}{h^2}$
(by lambert's cosine law)
or, $(r^2 + h^2)^{3/2} = (2h)^3 $ or $(r^2 + h^2)^{1/2} = 2h$
or $r^2 + h^2 = 4h^2$
$h = r / \sqrt{3}$

Multiple choice maths similarity areas of similar figures areas of similar triangles relations between the areas of triangles

Let $\displaystyle \Delta XYZ$ be right angle triangle with right angle at Z. Let $\displaystyle A _{X}$ denotes the area of the circle with diameter YZ. Let $\displaystyle A _{Y}$ denote the area of the circle with diameter XZ and let $\displaystyle A _{Z}$ denotes the area of the circle diameter XY. Which of the following relations is true?

  1. $\displaystyle A _{Z}=A _{X}+A _{Y}$
  2. $\displaystyle A _{Z}=A^{2} _{X}+A^{2} _{Y}$
  3. $\displaystyle A^{2} _{Z}=A^{2} _{X}+A^{2} _{Y}$
  4. $\displaystyle A^{2} _{Z}=A^{2} _{X}-A^{2} _{Y}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In $\triangle XYZ$, using Pythagoras theorem,
$XY^2 = XZ^2 + YZ^2$
$\pi XY^2 = \pi XZ^2 + \pi YZ^2$ (Multiply by $\pi$)
$A _z = A _x + A _y$

Multiple choice maths introduction to three dimensional geometry distance between two points in 3d distance between two points in space scalars and vectors

The circum radius of the triangle formed by the points $(0, 0, 0)$, $(0, 0, 12)$ and $(3, 4, 0)$ is

  1. $\sqrt{156}$
  2. $13$
  3. $\displaystyle \frac { 13 }{ 2 } $
  4. $8$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Center of the circum circle of a right angle triangle is on the mid of the hypotenuse. 

Hence the radius is the half of the length of the hypotenuse.
$\dfrac{\sqrt{3^2+4^2+12^2}}{2}=\dfrac{13}{2}$ 

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

From a point A which is at a distance of 10 cm from the center O of a circle of radius 6 cm, the pair of tangents AB and AC to the circle are drawn. Then the area of Quadrilateral ABOC is:

  1. $24 cm^{2}$
  2. $4 8cm^{2}$
  3. $96cm^{2}$
  4. $100cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since $\triangle ABO$ is congruent to $\triangle ACO$,  area of $ABOC$ is twice the area of $\triangle ABO$.

In $\triangle ABO, \ OA = 10 cm, \ OB = 6 cm$.
Since tangent is perpendicular to radius at the point of contact, by Pythagoras' theorem, we have 
$AB = \sqrt{OA^2 - OB^2} = 8 cm$
So, the area of $\triangle ABO$ is $\dfrac{1}{2}\times AB\times OB = 24 cm^2$
So, the area of $ABOC$ is $2\times 24 = 48 cm^2$.
So option B is the right answer.