3.5 The unit circle questions for IB Maths AA

67 questions on The unit circle, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
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	33	14	47
Paper 2	18	1	19
Paper 3HL only	0	1	1
Every paper	51	16	67

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 · AA-00079Paper 1 · 4 marks

Triangle $A B C$ has a right angle at $B$ , with $A B = 6$ cm and $A \hat{C} B = 30 °$ . The point $D$ lies on $[B C]$ such that $A \hat{D} B = 60 °$ .

Find the exact length of $[D C]$ .

4 marks

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Question 2 · AA-00086Paper 2 · 6 marks

In triangle $A B C$ , $A C = 9$ cm, $B C = 14$ cm and $A \hat{C} B$ is obtuse. The area of the triangle is $50$ cm $^{2}$ .

(a)
Find $A \hat{C} B$ .
3 marks
(b)
Find $A B$ .
3 marks

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Question 3 · AA-01557Paper 3 · HL only · 28 marks

This question asks you to investigate how regular polygons inscribed in a circle can be used to approximate $π$ , and how the approximation can be improved.

A regular polygon with $n$ sides, where $n \in Z$ and $n \geq 3$ , is drawn inside a circle of radius $1$ so that all its vertices lie on the circle. Let $A_{n}$ be the area of the polygon and let $p_{n}$ be half of its perimeter.

(a)
Show that $A_{n} = \frac{n}{2} \sin (\frac{2 π}{n})$ .
2 marks
(b)
Show that $p_{n} = n \sin (\frac{π}{n})$ .
2 marks
(c)
Find the exact values of $p_{6}$ , $A_{6}$ and $A_{12}$ .
3 marks
(d)
Use your GDC to find the values of $p_{24}$ and $A_{24}$ , giving your answers correct to five significant figures.
2 marks
(e)
By writing $x = \frac{1}{n}$ , use l'Hôpital's rule to show that $\lim_{n \to \infty} p_{n} = π$ .
3 marks
(f)
Find $\lim_{n \to \infty} A_{n}$ .
2 marks
(g)
Use l'Hôpital's rule to show that $\lim_{n \to \infty} n^{2} (π - p_{n}) = \frac{π^{3}}{6}$ .
5 marks
(h)
It can be shown that $\lim_{n \to \infty} n^{2} (π - A_{n}) = \frac{2 π^{3}}{3}$ .
Let $c_{n} = λ p_{n} + (1 - λ) A_{n}$ , where $λ \in R$ . Find the value of $λ$ for which $\lim_{n \to \infty} n^{2} (π - c_{n}) = 0$ .
4 marks
(i)
For this value of $λ$ , find the exact value of $c_{6}$ .
2 marks
(j)
With $λ$ equal to the value found in part (h), find the least value of $n$ for which $π - p_{n} < 10^{- 4}$ , and the least value of $n$ for which $π - c_{n} < 10^{- 4}$ .
3 marks

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Question 4 · AA-00080Paper 1 · 7 marks

In triangle $A B C$ , $A B = x$ cm, $A C = (x + 4)$ cm and $B \hat{A} C = 150 °$ . The area of the triangle is $15$ cm $^{2}$ .

(a)
Find the value of $x$ .
4 marks
(b)
Find $B C$ , giving your answer in the form $\sqrt{p + q \sqrt{3}}$ , where $p, q \in Z^{+}$ .
3 marks

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Question 5 · AA-00091Paper 2 · 15 marks

A robotic arm has two straight sections, $[S E]$ and $[E H]$ , joined at E. The section $[S E]$ has length $0.8$ m and the section $[E H]$ has length $0.6$ m. The point S is fixed. The angle $S \hat{E} H = θ$ , where $0 ° < θ < 180 °$ , can be changed.

(a)
Show that ${S H}^{2} = 1 - 0.96 \cos θ$ .
2 marks
(b)
Find $S H$ when $θ = 110 °$ .
2 marks
(c)
For $θ = 110 °$ , find $E \hat{S} H$ .
3 marks
(d)
Find the value of $θ$ for which $S H = 1.2$ m.
3 marks
(e)
The area of triangle $S E H$ is $0.2$ m $^{2}$ . Find the two possible values of $θ$ .
3 marks
(f)
For the larger value of $θ$ found in part (e), find $S H$ .
2 marks

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Question 6 · AA-00082Paper 1 · 6 marks

In triangle $A B C$ , $A B = 5$ , $A C = 7$ and $A \hat{B} C = 120 °$ .

(a)
Find $B C$ .
4 marks
(b)
Find the exact area of triangle $A B C$ .
2 marks

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