3.4 Arcs and sectors questions for IB Maths AI

31 questions on Arcs and sectors, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Applications and Interpretation
Part of 3 Geometry and trigonometry
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	14	2	16
Paper 2	12	3	15
Every paper	26	5	31

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AI-00505Paper 1 · HL only · 7 marks

A robot arm is pivoted at the origin of an Argand diagram, where $1$ unit represents $1$ cm. The tip of the arm is at the point $T$ , which represents $30 + 40 i$ .

The arm then rotates $50 °$ anticlockwise about the origin.

(a)
Write $30 + 40 i$ in the form $r e^{i θ}$ .
2 marks
(b)
By multiplying by a suitable complex number, find the new position of the tip of the arm. Give your answer in the form $a + b i$ .
3 marks
(c)
Find the straight-line distance between the initial and final positions of the tip of the arm.
2 marks

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Question 2 · AI-00228Paper 2 · 16 marks

Archaeologists excavating a site find three post holes P(1, 2), Q(8, 9) and R(9, 2) from a circular hut. Distances are in metres. They believe all three post holes lie on the circular wall of the hut.

Coordinate grid from 0 to 10 on both axes, 1 unit = 1 m. Post holes P at (1, 2), Q at (8, 9) and R at (9, 2).

(a)
Write down the equation of the perpendicular bisector of [PR].
1 mark
(b)
Find the equation of the perpendicular bisector of [PQ].
4 marks
(c)
Hence find the coordinates of C, the centre of the hut.
2 marks
(d)
Find the radius of the hut.
2 marks
(e)
Find the floor area of the hut.
2 marks
(f)
Find the size of angle PCR.
3 marks
(g)
The wall between P and R that does not pass through Q has been destroyed. Find the length of this section of wall.
2 marks

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Question 3 · AI-00712Paper 1 · 6 marks

A circular pizza has a diameter of 32 cm. It is cut into 8 equal sector-shaped slices.

(a)
Write down the angle at the centre of one slice.
1 mark
(b)
Find the area of the top of one slice.
2 marks
(c)
Find the perimeter of one slice.
3 marks

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Question 4 · AI-00350Paper 2 · 14 marks

An outdoor theatre is built as an amphitheatre. Seen from above, the stage is at O and the rows of benches (tiers) are arcs of circles centred at O, each subtending an angle of 140° at O.

The first tier has radius 12 m, and the radius of each tier is 0.9 m greater than that of the tier in front of it.

Plan view of an amphitheatre: a sector with centre O and angle 140°. The first tier is an arc of radius 12 m from O, and each further tier is an arc 0.9 m further out. Tiers 1 to 3 are drawn, and further tiers continue outwards. — Diagram not to scale.

(a)
Show that the length of the arc of the first tier is 29.3 m, correct to three significant figures.
2 marks
(b)
The arc lengths of the tiers form an arithmetic sequence. Find the common difference.
2 marks
(c)
Find the length of the arc of the 20th tier.
2 marks
(d)
Each spectator needs 0.55 m of bench. Find the maximum number of spectators who can sit on the 20th tier.
2 marks
(e)
Find the total length of the arcs of the first 20 tiers. Give your answer to the nearest metre.
2 marks
(f)
The architect, Chiara, wants the amphitheatre to seat at least 3000 spectators. She assumes that the whole length of every tier can be used.
Find the least number of tiers needed.
3 marks
(g)
Explain why, in practice, more tiers than your answer to part (f) may be needed.
1 mark

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Question 5 · AI-00713Paper 1 · 7 marks

A flower bed in a park is in the shape of a sector of a circle with radius 4.5 m and angle 110°.

(a)
Find the length of the curved edge of the flower bed.
2 marks
(b)
A metal edging strip is placed around the whole perimeter of the flower bed. The edging costs 12 euros (EUR) per metre. Find the cost of the edging. Give your answer correct to two decimal places.
3 marks
(c)
Find the area of the flower bed.
2 marks

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Question 6 · AI-00816Paper 2 · HL only · 14 marks

Lucas pulls a playground swing back and releases it. The seat of the swing hangs 2.4 m from its pivot and moves along an arc of a circle. On its first swing the chain turns through an angle of 75°. On each swing after that, the angle turned through is 94% of the angle of the previous swing.

(a)
Show that the length of the arc travelled by the seat on the first swing is $π$ metres.
2 marks
(b)
Find the angle turned through on the fifth swing.
2 marks
(c)
Find the total distance travelled by the seat in the first 20 swings.
3 marks
(d)
Find the total distance the seat would travel if it continued swinging indefinitely.
2 marks
(e)
Find the number of the first swing on which the chain turns through less than 10°.
3 marks
(f)
Find the percentage of the distance in part (d) that has already been travelled by the end of the swing found in part (e).
2 marks

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