A sector has radius $12\text{ cm}$ and central angle $45^\circ$. Calculate the arc length to 3 s.f.

1. Formula : $l = \frac{\theta}{360} \times 2\pi r$. 2. Substitute : $l = \frac{45}{360} \times 24\pi = 3\pi$. 3. State : $3\pi = \mathbf{9.42\text{ cm}}$.

A sprinkler covers a sector of radius $8.5\text{ m}$ sweeping $140^\circ$. Calculate the area to 3 s.f.

1. Formula : $A = \frac{\theta}{360}\pi r^2$. 2. Substitute : $A = \frac{140}{360}\pi(8.5)^2$. 3. Compute : $A = 88.27\ldots$ 4. State : $\mathbf{88.3\text{ m}^2}$.

A sector has radius $10\text{ cm}$ and area $60\text{ cm}^2$. Find the total perimeter.

1. Find $\theta$ : $60 = \frac{\theta}{360}\pi(100) \implies \theta \approx 68.75^\circ$. 2. Arc length : $l = \frac{68.75}{360}(20\pi) = 12\text{ cm}$. 3. Perimeter : arc $+ 2r = 12 + 20$. 4. State : $\mathbf{32.0\text{ cm}}$.

A sector of a circle has a radius of $5 \text{ cm}$ and an area of $25 \text{ cm}^2$. Calculate the size of the central angle, $\theta$, in degrees. [2 marks] Hence, calculate the arc length of the sector. [2 marks]

$\dfrac{\theta}{360} \times \pi (5)^2 = 25$ M1 $\theta = \dfrac{360}{\pi} = 114.6^\circ$ (114.59…) A1 Arc length $= \dfrac{\theta}{360} \times 2\pi(5)$ M1 $= 10\text{ cm}$ A1

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IB Maths AI SL · Unit 3: Geometry and Trigonometry

IB Maths AI SL Arcs and Sectors Questions

Updated October 2026

Exam-style IB Maths AI SL arcs and sectors questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

13 questions
Paper 1: 4
Paper 2: 9
2 easy
11 medium
3 worked examples

Practise Arcs and Sectors questions →

Key formulas

Area of a sector: $A = π r^{2} \cdot \frac{θ}{360^{\circ}}$

In the same notation as the IB formula booklet. All AI SL formulas →

Explained, with a worked example: Arc length · Area of a sector

Arcs and Sectors worked examples

Worked example 1: Arc length in degrees · easy

A sector has radius $12 cm$ and central angle $45^{\circ}$ . Calculate the arc length to 3 s.f.

Solution

3 steps

Examiner tip: Arc length uses

2 π r

; sector area uses

π r^{2}

. Different formulas.

Worked example 2: Sector area — sprinkler coverage · medium

A sprinkler covers a sector of radius $8.5 m$ sweeping $140^{\circ}$ . Calculate the area to 3 s.f.

Solution

4 steps

Examiner tip: AI SL uses degrees for sectors; radian versions belong to AA.

Worked example 3: Perimeter of a sector from its area · hard

A sector has radius $10 cm$ and area $60 {cm}^{2}$ . Find the total perimeter.

Solution

4 steps

Examiner tip: Perimeter of a sector = arc + TWO radii. Forgetting

+ 2 r

is the classic trap.

Try these IB Maths AI SL arcs and sectors questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 3 marks · Paper 1

A sector of a circle has a radius of $10$ cm and a central angle of $45^{\circ}$ . Calculate the exact length of the arc.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 2

A sector of a circle has a radius of $9 cm$ and an area of $45 {cm}^{2}$ .

Calculate the size of the central angle, $θ$ , in degrees.
Hence, calculate the perimeter of the sector.

Attempt it and see the mark scheme →

Question 3 · medium · 4 marks · Paper 2

A sector of a circle has a radius of $5 cm$ and an area of $25 {cm}^{2}$ .

Calculate the size of the central angle, $θ$ , in degrees. [2 marks]
Hence, calculate the arc length of the sector. [2 marks]

Full worked solution Free, no sign-in

$\frac{θ}{360} \times π (5)^{2} = 25$ M1
$θ = \frac{360}{π} = {114.6}^{\circ}$ (114.59…) A1
Arc length $= \frac{θ}{360} \times 2 π (5)$ M1
$= 10 cm$ A1

Every question here has a full worked solution. See them all →

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FAQ

How many IB Maths AI SL arcs and sectors questions are there?

There are 13 exam-style arcs and sectors questions in the AI SL question bank (Paper 1: 4 · Paper 2: 9), graded 11 medium, 2 easy. Every question has a full IB-style mark scheme (M, A and R marks).

Is arcs and sectors on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 4 · Paper 2: 9. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 3 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

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Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Free ready-to-teach lesson (slides and teacher notes): Arc length & sector area.

IB Maths AI SL Arcs and Sectors Questions

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Key formulas

Arcs and Sectors worked examples

Worked example 1: Arc length in degrees · easy

Solution

Worked example 2: Sector area — sprinkler coverage · medium

Solution

Worked example 3: Perimeter of a sector from its area · hard

Solution

Try these IB Maths AI SL arcs and sectors questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 5 marks · Paper 2

Question 3 · medium · 4 marks · Paper 2

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Revise arcs and sectors for the exam

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