A chord of length $18$ cm is drawn across a circle of radius $12$ cm. Find the central angle $\theta$ using the cosine rule, and the area of the minor segment.

1. Cosine Rule : $18^2 = 12^2 + 12^2 - 2(12)(12)\cos\theta$. 2. Simplify : $324 = 288 - 288\cos\theta$, so $\cos\theta = -0.125$. 3. Inverse cosine (Radian mode): $\theta = 1.6961\ldots$ rad. 4. Segment area $= \frac{1}{2}r^2(\theta - \sin\theta) = 72(1.6961\ldots - 0.9921\ldots) = \mathbf{50.7\text{ cm}^2}$.

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IB Maths AA SL · Unit 3: Geometry and Trigonometry

IB Maths AA SL Radians Questions

Updated October 2026

Exam-style IB Maths AA SL radians 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.

44 questions
Paper 1: 29
Paper 2: 15
10 easy
20 medium
10 hard
4 very hard
3 worked examples

Practise Radians questions →

What's examined in AA SL radians

The question bank covers these radians question types (number of questions in brackets):

Radial Problem Solving (19)
Sector Arc & Area (14)
Segments & Chords (11)

Key formulas

Arc length (radians): $ℓ = r θ$
Area of a sector (radians): $A = \frac{1}{2} r^{2} θ$

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

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

Where students lose marks on radians

Examiner's noteGDC in the wrong angle mode, and degrees in radian formulas

Where marks are lost: Reported in four of the five sessions: kinematics and trigonometric models worked in degree mode, the sector-area formula used with the angle in degrees, the GDC's simultaneous-equation solver used in the wrong mode, and a bearing question done in radians.

Examiners flagged this in 3 exam series (May 2024, May 2025, November 2025). Summarised in our own words from IB subject reports, IB Programme Resource Centre. Read the full note →

Radians worked examples

Worked example 1: Finding angle from perimeter · easy

A sector of a circle with radius $6$ cm has a total perimeter of $30$ cm. Find the exact angle $θ$ in radians, and the exact area.

Solution

3 steps

Examiner tip: "Perimeter of a sector" is arc length PLUS the two radii — don't forget the

2 r

straight edges!

Worked example 2: Area of a minor segment · medium

A chord of length $18$ cm is drawn across a circle of radius $12$ cm. Find the central angle $θ$ using the cosine rule, and the area of the minor segment.

Solution

4 steps

Examiner tip: The segment formula

\frac{1}{2} r^{2} (θ - \sin θ)

requires

θ

in RADIANS. Substituting degrees produces nonsense.

Worked example 3: Sector prisms and surface area · hard

A prism with sector cross-section (central angle $1.05$ rad) has height $4$ mm and volume $412 {mm}^{3}$ . Find its total surface area.

Solution

6 steps

Examiner tip: When a "slice of cake" prism is unpacked into its net, students often forget the two flat rectangular sides exposed when the slice was cut out.

Try these IB Maths AA SL radians 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 · 5 marks · Paper 2

A lawn sprinkler sprays water over a lawn covering an arc of $1.8$ radians with a maximum spray distance of $r$ metres. The lawn sprinkler perfectly waters a sector of area $20 m^{2}$ of the lawn.

Calculate the value of the maximum spray distance, $r$ .
Calculate the length of the outer arc watered by the sprinkler.

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 1

Consider a sector of a circle with radius $R cm$ and an angle of $θ$ radians at the centre. The area of the sector (in ${cm}^{2}$ ) is numerically equal to exactly three times the length of its arc (in $cm$ ). Find the exact value of $R$ .

Full worked solution Free, no sign-in

Set up the expressions for area and arc length.
Area $= \frac{1}{2} R^{2} θ$ . (A1)
Arc length $= R θ$ . (A1)
Equating the area to three times the arc length:
$\frac{1}{2} R^{2} θ = 3 (R θ)$ . (M1)
Since $θ > 0$ and $R > 0$ , we can divide both sides by $R θ$ : (M1)
$\frac{1}{2} R = 3 ⟹ R = 6$ . A1

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

Question 3 · hard · 6 marks · Paper 2

A games design company produces solid plastic game pieces in the form of a right prism. The uniform cross-section of the prism is the sector of a circle. The angle at the centre of the sector is $1.05$ radians, and the uniform height of the game piece is $4 mm$ .

Given that the total volume of the game piece is $412 {mm}^{3}$ , calculate the radius of the sector cross-section.
Hence, calculate the total surface area of the plastic game piece in ${mm}^{2}$ .

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AA SL radians questions are there?

There are 44 exam-style radians questions in the AA SL question bank (Paper 1: 29 · Paper 2: 15), graded 10 easy, 20 medium, 10 hard, 4 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is radians on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 29 · Paper 2: 15. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.

Where can I get the mark schemes?

Open the AA 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.

What else is free here

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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Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Radian measure · Arc length & sector area.

IB Maths AA SL Radians Questions

Try it first

What's examined in AA SL radians

Key formulas

Where students lose marks on radians

Radians worked examples

Worked example 1: Finding angle from perimeter · easy

Solution

Worked example 2: Area of a minor segment · medium

Solution

Worked example 3: Sector prisms and surface area · hard

Solution

Try these IB Maths AA SL radians questions

Question 1 · easy · 5 marks · Paper 2

Question 2 · medium · 5 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Revise radians for the exam

FAQ

What else is free here

Teaching this topic?

More AA SL Unit 3 topics