RevisionDojo

SL 3.4—Circle, radians, arcs, sectors - IB Questionbank | RevisionDojo

Practice SL 3.4—Circle, radians, arcs, sectors with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

Express the following angles in radian, give your answer in terms of $\pi$ .

a. $10^{\circ}$ b. $18^{\circ}$ c. $270^{\circ}$

[3]

2.Express the following angles in degrees.

a. $\frac{\pi}{8}$ b. $\frac{\pi}{3}$ c. $3 \pi$

[3]

Question 2

SLPaper 1

For the given diagram, point A represents the angle of $30^{\circ}$ on the unit circle.

Determine the values of the angle at point A within the following intervals in degrees and in radians :

$-720^{\circ}<\theta<-360^{\circ}$ or $-4 \pi<\theta<-2 \pi$

[1]

$-360^{\circ}<\theta<0^{\circ}$ or $-2 \pi<\theta<0$

[1]

$360^{\circ}<\theta<720^{\circ}$ or $2 \pi<\theta<4 \pi$

[1]

$720^{\circ}<\theta<1080^{\circ}$ or $4 \pi<\theta<4 \pi$

[1]

Question 3

SLPaper 2

Given the circle at center $O$ whose radius is 9 cm and size of angle AOB is 1.5 radian.

The minor arc is shaded and the major arc is unshaded.

Find the length of the minor arc AB and the major arc AB .

[2]

Find the area of the minor sector and the major sector.

[2]

find the perimeter of the minor sector and the major sector.

[2]

Question 4

SLPaper 2

In the given figure, the sector of a circle is shown having radius $r \mathrm{~cm}$ and angle $\theta$ at the centre. The perimeter of the sector is 20 cm .

Show that the angle $\theta$ is $\frac{20-2 r}{r}$

[3]

The area of the sector is $25 \mathrm{~cm}^{2}$ . Find the value of $r$ .

[3]

Question 5

SLPaper 1

For the given diagram, point A represents the angle of $30^{\circ}$ on the unit circle.

Find representation of the points $\mathrm{B}, \mathrm{C}$ and D in degrees within the interval $\left[0^{\circ}, 360^{\circ}\right]$ and in radians within the interval $[0,2 \pi]$ .

[3]

For the given diagram, point C represents the angle of $220^{\circ}$ on the unit circle. Find representation of the points $\mathrm{A}, \mathrm{B}$ and D in degrees within the interval $\left[0^{\circ}, 360^{\circ}\right]$ and in radians within the interval $[0,2 \pi]$ .

[3]

Question 6

SLPaper 2

In the given figure, the centre of a circle is at O with radius 1 and angle AOB is $\theta$ where $\theta \neq 0$ . The area of $\triangle \mathrm{AOB}$ is three times the angle of the shaded portion.

show that $3 \theta=4 \sin \theta$ .

[4]

To find the value of $\theta$ .

[2]

Question 7

SLPaper 2

Two observation towers, $A$ and $B$ , stand on level ground 250 m apart.
Tower $A$ is 40 m high, tower $B$ is 70 m high.
A drone $D$ is observed simultaneously from the tops of both towers.
From $A$ , the angle of elevation of $D$ is $25^\circ$ ; from $B$ , the angle of elevation is $38^\circ$ .
The horizontal bearing of $B$ from $A$ is $090^\circ$ .
The bearing of the drone from $A$ is $042^\circ$ , and from $B$ is $330^\circ$ .

Draw a clearly labelled 3-D diagram showing the towers, the ground separation, the bearings, and the line of sight from each tower to $D$ .

[2]

Let the point vertically below the drone be $P$ on the ground.
Find the horizontal distances $AP$ and $BP$ .

[3]

Hence find the height of the drone above the ground and its straight-line 3-D distance from the top of tower $A$ .

[3]

The drone now circles tower $A$ at constant height $106.3$ m, describing a horizontal circle of radius 200 m.
A camera on $A$ can rotate through a maximum of $160^\circ$ before needing to reset.
Determine the length of visible path of the drone from $A$ before the camera must reset, giving your answer to the nearest metre.

[2]

During this circular motion, the drone begins a climb such that its height increases proportionally to the square of the horizontal angle turned:
$h(\theta) = 106.3 + 2.5\theta^2 \quad (0 \le \theta \le 2.7925),$ where $\theta$ is in radians.
Find, correct to three significant figures,

the total 3-D distance flown by the drone over this visible path, and
the greatest angle of elevation of the drone from the top of tower A.

[5]

Question 8

SLPaper 2

In the given figure, O is the centre of the circle having radius 5 cm . Points A and B are on the circle such that angle AOB is 0.8 radians. The point N lies on line OB such that AN is perpendicular to OB .

Find the area of the shaded region.

[4]

Question 9

SLPaper 2

Given the circle whose radius is 5 cm and size of angle AOB is $65^{\circ}$ .

Find the length of the minor arc AB and the major arc AB .

[4]

Find the area of the shaded region.

[2]

Question 10

SLPaper 2

The given diagram shows a circle with center O and radius 12 cm . The chord AB subtends at an angle of $75^{\circ}$ at the centre. The tangents to the circle at A and B meet at P .

Use the cosine rule to show the length of AB is $12 \sqrt{2\left(1-\cos 75^{\circ}\right)}$

[3]

Find the length of BP .

[3]

Find the area of the triangle OBP .

[3]

Find the area of the triangle ABP .

[3]

Find the area of the sector OAB .

[2]

Find the area of the shaded region.

[2]

SL 3.4—Circle, radians, arcs, sectors Questionbank