IB Maths AA SL · Unit 3: Geometry and Trigonometry
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
The question bank covers these radians question types (number of questions in brackets):
A sector of a circle with radius $6$ cm has a total perimeter of $30$ cm. Find the exact angle $\theta$ in radians, and the exact area.
1. Perimeter $= 2r + r\theta$: $30 = 12 + 6\theta$.
2. Solve: $\theta = 3$ radians.
3. Area $= \frac{1}{2}r^2\theta = \frac{1}{2}(36)(3) = \mathbf{54\text{ cm}^2}$.
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 $= \frac{1}{2}r^2(\theta - \sin\theta) = 72(1.6961\ldots - 0.9921\ldots) = \mathbf{50.7\text{ cm}^2}$.
A prism with sector cross-section (central angle $1.05$ rad) has height $4$ mm and volume $412\text{ mm}^3$. Find its total surface area.
1. Volume: $412 = \text{Sector Area} \times 4$, so Sector Area $= 103\text{ mm}^2$.
2. Find $r$: $103 = \frac{1}{2}r^2(1.05) \implies r^2 = 196.19\ldots \implies r = 14.006\ldots$ mm.
3. Arc length: $14.006\ldots \times 1.05 = 14.707\ldots$
4. Curved surface: $14.707\ldots \times 4 = 58.828\ldots$
5. Two rectangular sides: $2(14.006\ldots \times 4) = 112.05\ldots$
6. TSA: $2(103) + 58.828\ldots + 112.05\ldots = \mathbf{377\text{ mm}^2}$.
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
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 \(20\text{ m}^2\) of the lawn.
Calculate the value of the maximum spray distance, \(r\).
Calculate the length of the outer arc watered by the sprinkler.
Consider a sector of a circle with radius \(R\text{ cm}\) and an angle of \(\theta\) radians at the centre. The area of the sector (in \(\text{cm}^2\)) is numerically equal to exactly three times the length of its arc (in \(\text{cm}\)). Find the exact value of \(R\).
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 \(4\text{ mm}\).
Given that the total volume of the game piece is \(412\text{ mm}^3\), calculate the radius of the sector cross-section.
Hence, calculate the total surface area of the plastic game piece in \(\text{mm}^2\).
There are 29 exam-style radians questions in the AA SL question bank (Paper 1: 18 · Paper 2: 11), graded 5 easy, 12 medium, 7 hard, 3 very hard, 2 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Both. In the bank, Paper 1: 18 · Paper 2: 11. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
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.