SL 3.4—Circle, radians, arcs, sectors Questionbank

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 2

A research drone is tethered by two cables, $AP$ and $BP$ , attached to points $A$ and $B$ on level ground. The points $A$ and $B$ are 80 m apart. The drone $P$ is directly above the point $P_0$ on the ground. In the horizontal triangle $ABP_0$ , the angle at $P_0$ between the lines $P_0A$ and $P_0B$ is $60^\circ$ .

At a certain instant, the angle of elevation of the drone from $A$ is $28^\circ$ and from $B$ is $35^\circ$ .

Draw a fully-labelled 3-D diagram showing $A, B, P_0, P$ and all known angles.

[2]

Let the horizontal distances $AP_0 = x$ m and $BP_0 = y$ m. Show that $x \tan 28^\circ = y \tan 35^\circ.$

[2]

Using the cosine rule in triangle $ABP_0$ , show that $80^2 = x^2 + y^2 - 2xy \cos 60^\circ.$

[2]

Use your results from part 2 and 3 to find the height $h$ of the drone above the ground. Give your answer to the nearest metre.

[3]

The drone then moves horizontally so that its projection $P_0$ traces out a circular path of radius 80 m centred at $A$ . Calculate the angle swept by $P_0$ in radians and the length of the arc of this path when the bearing of $P_0$ from $A$ changes from $050^\circ$ to $130^\circ$ .

[3]

The tension in each cable is proportional to its length. If the tension in $AP$ is $T_A = k|AP|$ and in $BP$ is $T_B = k|BP|$ , find the ratio $\dfrac{T_A}{T_B}$ at this instant.

[3]

Question 2

SLPaper 2

A city planner is designing a circular flower bed in a public park. A sector of this circular flower bed will be planted with seasonal flowers, while the rest will have perennial plants. The area of the sector planted with seasonal flowers is $87.3 \, \text{m}^2$ with a central angle of $1.3$ radians.

Calculate the radius of the circular flower bed.

[3]

Determine the perimeter of the sector that will be used for seasonal flowers.

[3]

Question 3

SLPaper 2

A rescue helicopter $P$ hovers above a mountain valley. It is connected by two winch cables to huts $A$ and $B$ on level ground. The distance between the huts is $90 \text{ m}$ . The helicopter is vertically above point $P_0$ , and in the horizontal triangle $ABP_0$ , $\angle AP_0B = 70^\circ$ .

The angle of elevation of the helicopter is $32^\circ$ from $A$ and $45^\circ$ from $B$ .

Sketch a fully-labelled 3-D diagram showing $A, B, P_0, P$ and all given angles.

[2]

Let $AP_0=x$ m and $BP_0=y$ m. Show that $x\tan32^\circ = y\tan45^\circ.$

[2]

Using the cosine rule in triangle $ABP_0$ , write down an equation relating $x$ and $y$ .

[2]

Use your results to find the height $h$ of the helicopter above the ground, correct to the nearest metre.

[2]

The point $P_0$ later moves horizontally around a circular path of radius $90 \text{ m}$ centred at $A$ . Find the length of the arc, in metres and radians, when its bearing from $A$ changes from $010^\circ$ to $100^\circ$ .

[3]

If tensions are $T_A = k|AP|$ and $T_B = k|BP|$ , find $\dfrac{T_A}{T_B}$ .

[3]

Question 4

SLPaper 2

A hot-air balloon $P$ is anchored by two ropes to points $A$ and $B$ on level ground. The distance $AB$ is 75 m. The balloon is vertically above $P_0$ , and $\angle AP_0B = 60^\circ$ in the horizontal triangle $ABP_0$ .

The angles of elevation of the balloon are $28^\circ$ from $A$ and $37^\circ$ from $B$ .

Draw a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all the given angles.

[2]

Let $AP_0=x$ m and $BP_0=y$ m. Show that

x\tan28^\circ = y\tan37^\circ.

[2]

Using the cosine rule in triangle $ABP_0$ , write an equation relating $x$ and $y$ .

[2]

Find the height $h$ of the balloon, to the nearest metre.

[2]

The projection $P_0$ moves horizontally on a circle of radius $75\text{ m}$ centered at $A$ . Find the angle subtended by the arc in radians and the length of the arc in metres when the bearing of $P_0$ from $A$ changes from $045^\circ$ to $115^\circ$ .

[3]

If $T_A = k|AP|$ and $T_B = k|BP|$ , find $\dfrac{T_A}{T_B}$ .

[3]

Question 5

SLPaper 1

Express the following angles in radians, giving your answers in terms of $\pi$ . a. $10^{\circ}$ b. $18^{\circ}$ c. $270^{\circ}$

[3]

Express the following angles in degrees. a. $\frac{\pi}{8}$ b. $\frac{\pi}{3}$ c. $3 \pi$

[3]

Question 6

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 $4 \pi<\theta<6 \pi$

[1]

Question 7

SLPaper 2

Consider a circle with center $O$ and radius $9 \text{ cm}$ . The size of angle $AOB$ is $1.5$ radians.

The minor sector is shaded.

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

[2]

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

[2]

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

[2]

Question 8

SLPaper 2

Convert the following angles from degrees to radians, giving each answer to three significant figures.

Convert $28^\circ$ to radians.

[2]

Convert $36^\circ$ to radians.

[2]

Convert $67^\circ$ to radians.

[2]

Question 9

SLPaper 2

A satellite dish $D$ is positioned vertically above a point $D_0$ on the ground. Two control posts, $C_1$ and $C_2$ , are 40 m apart. The bearing of $C_2$ from $C_1$ is $090^\circ$ . The bearing of the dish $D_0$ from $C_1$ is $010^\circ$ . At a certain instant, the angle of elevation of the dish from $C_1$ is $60^\circ$ and from $C_2$ is $45^\circ$ .

Draw a fully-labelled 3D diagram showing $C_1,C_2,D_0,D$ and all known angles.

[2]

Let the horizontal distances $C_1 D_0 = x$ m and $C_2 D_0 = y$ m. Show that

x\tan60^\circ = y\tan45^\circ .

[2]

Using the cosine rule in triangle $C_1C_2D_0$ , express $y$ in terms of $x$ and known constants.

[3]

Combine your results from parts 2 and 3 to find the height $h$ of the dish. Give your answer to the nearest tenth of a metre.

[3]

The dish's projection $D_0$ describes a circular path of radius $x$ m. The path subtends an angle of $2.5$ radians. Calculate the arc length in metres.

[3]

Find the ratio of the horizontal distance from $C_2$ to the 3D distance $|C_2 D|$ at this instant.

[3]

Question 10

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 $20 \text{ cm}$ .

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]