A cube has side length $a$. Prove that the exact angle $\theta$ between its main interior diagonal and its square base satisfies $\sin\theta = \frac{1}{\sqrt{3}}$.

1. Base diagonal via Pythagoras: $\sqrt{a^2+a^2} = a\sqrt{2}$. 2. Main diagonal via Pythagoras again: $\sqrt{(a\sqrt{2})^2 + a^2} = \sqrt{3a^2} = a\sqrt{3}$. 3. Sine ratio : opposite is the vertical edge $a$, hypotenuse is $a\sqrt{3}$. 4. Simplify : $\sin\theta = \frac{a}{a\sqrt{3}} = \mathbf{\frac{1}{\sqrt{3}}}$.

A solid object is constructed by placing a solid hemisphere of radius $r$ precisely on top of a solid cylinder of radius $r$ and height $h$ . Given that the volume of the hemisphere is equal to the volume of the cylinder, express $h$ in terms of $r$ . Show that the total surface area of the combined solid is exactly $\frac{13}{3}\pi r^2$ .

Volume of hemisphere $= \frac{2}{3}\pi r^3$ and volume of cylinder $= \pi r^2 h$ . Equate the volumes: $\frac{2}{3}\pi r^3 = \pi r^2 h$ . M1A1 Divide by $\pi r^2$ : $h = \frac{2}{3}r$ . A1 Total surface area = curved area of hemisphere + curved area of cylinder + flat base: $SA = 2\pi r^2 + 2\pi r h + \pi r^2$ . M1A1 Substitute $h = \frac{2}{3}r$ : $SA = 3\pi r^2 + 2\pi r \left(\frac{2}{3}r\right) = 3\pi r^2 + \frac{4}{3}\pi r^2$ . A1 $SA = \frac{9}{3}\pi r^2 + \frac{4}{3}\pi r^2 = \frac{13}{3}\pi r^2$ . AG

Geometry and Right Angled Trigonometry Questions

What you need to know

sin²θ + cos²θ = 1, plus the double-angle formulas. SL AA Paper 1 asks you to prove or simplify — never to memorise the derivation, but to APPLY it fluently. Trigonometric identities — Pythagorean and double-angle overview →

The unified triangle toolkit. Sine rule when you have angle-side pairs, cosine rule when you have SSS or SAS. Area = ½ab·sin(C) closes the trilogy. Sine rule, cosine rule, and area of a triangle overview →

What's examined in AA SL geometry and right angled trigonometry

The question bank covers these geometry and right angled trigonometry question types (number of questions in brackets):

Geometric Mensuration and Ratios (31)
Lengths and Angles in 3D (16)
3D Coordinate Geometry (15)

Key formulas

Double-angle: cosine: $\cos 2 θ = \cos^{2} θ - \sin^{2} θ = 2 \cos^{2} θ - 1 = 1 - 2 \sin^{2} θ$
Double-angle: sine: $\sin 2 θ = 2 \sin θ \cos θ$
Tangent from sine & cosine: $\tan θ = \frac{\sin θ}{\cos θ}$
Cosine rule: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$
Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Area of a triangle: $A = \frac{1}{2} a b \sin C$
Pythagorean identity: $\sin^{2} θ + \cos^{2} θ = 1$

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

Explained, with a worked example: Pythagoras' theorem · SOH CAH TOA · Volume of a cone · Volume of a sphere

Geometry and Right Angled Trigonometry worked examples

Worked example 1: Distance and midpoint in 3D · easy

Points $A (1, - 2, 3)$ and $B (5, 4, - 1)$ are located in a 3D Cartesian coordinate system. Find the exact distance between $A$ and $B$ , and the exact coordinates of their midpoint $M$ .

Solution

5 steps

Examiner tip: When calculating distance, always use brackets around negative numbers before squaring in your GDC; typing

- 4^{2}

instead of

(- 4)^{2}

gives

- 16

instead of

+ 16

.

Worked example 2: Slant height and surface area of a cone · medium

A right circular cone has a base radius of $r$ and a perpendicular height of $2 \sqrt{2} r$ . Find an exact expression for the slant height $l$ in terms of $r$ , and hence the exact ratio of the cone's curved surface area to its flat base area.

Solution

6 steps

Examiner tip: When squaring surd expressions like

2 \sqrt{2} r

, the square applies to every factor:

(2 \sqrt{2} r)^{2} = 4 \cdot 2 \cdot r^{2} = 8 r^{2}

.

Worked example 3: Angle between a 3D diagonal and a face · hard

A cube has side length $a$ . Prove that the exact angle $θ$ between its main interior diagonal and its square base satisfies $\sin θ = \frac{1}{\sqrt{3}}$ .

Solution

4 steps

Examiner tip: Always draw a separate flat 2D right-angled triangle for the cross-section, labelling "Opposite" and "Adjacent" before applying SOH CAH TOA.

Try these IB Maths AA SL geometry and right angled trigonometry 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

Find the exact length of the line segment joining the 3D coordinates $A (1, 2, 3)$ and $B (4, - 2, 5)$ .

Attempt it and see the mark scheme →

Question 2 · medium · 6 marks · Paper 1

A solid object is constructed by placing a solid hemisphere of radius $r$ precisely on top of a solid cylinder of radius $r$ and height $h$ .

Given that the volume of the hemisphere is equal to the volume of the cylinder, express $h$ in terms of $r$ .
Show that the total surface area of the combined solid is exactly $\frac{13}{3} π r^{2}$ .

Full worked solution Free, no sign-in

Volume of hemisphere $= \frac{2}{3} π r^{3}$ and volume of cylinder $= π r^{2} h$ .
Equate the volumes: $\frac{2}{3} π r^{3} = π r^{2} h$ . M1A1
Divide by $π r^{2}$ : $h = \frac{2}{3} r$ . A1
Total surface area = curved area of hemisphere + curved area of cylinder + flat base: $S A = 2 π r^{2} + 2 π r h + π r^{2}$ . M1A1
Substitute $h = \frac{2}{3} r$ : $S A = 3 π r^{2} + 2 π r (\frac{2}{3} r) = 3 π r^{2} + \frac{4}{3} π r^{2}$ . A1
$S A = \frac{9}{3} π r^{2} + \frac{4}{3} π r^{2} = \frac{13}{3} π r^{2}$ . AG

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

Question 3 · hard · 6 marks · Paper 2

A model building is created in the shape of a rectangular-based right pyramid $A B C D E$ . The base $A B C D$ is a rectangle with $A B = 4.6 cm$ and $B C = 7.2 cm$ . The apex $E$ is directly above the centre of the base, and each of the four slant edges $[A E]$ , $[B E]$ , $[C E]$ and $[D E]$ has length $8.3 cm$ .

Calculate the vertical height of the pyramid.
Calculate the volume of the model.
Find the angle that the triangular face $A B E$ (with base $4.6 cm$ ) makes with the rectangular base $A B C D$ .

Attempt it and see the mark scheme →

Free account — sign in to download Topic PDF: 7 questions with worked answers · 4 pages · 247 KB

All 62 geometry and right angled trigonometry questions with mark schemes →

Revise geometry and right angled trigonometry for the exam

FAQ

How many IB Maths AA SL geometry and right angled trigonometry questions are there?

There are 62 exam-style geometry and right angled trigonometry questions in the AA SL question bank (Paper 1: 33 · Paper 2: 29), graded 14 easy, 25 medium, 17 hard, 6 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is geometry and right angled trigonometry on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 33 · Paper 2: 29. 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.

Teaching this topic?

Free ready-to-teach lessons (slides and teacher notes): Distance & midpoint in 3D · Right-angle trig (SOH CAH TOA) · Angles of elevation & depression · Volume & surface area of 3D solids.

IB Maths AA SL Geometry and Right Angled Trigonometry Questions

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What you need to know

What's examined in AA SL geometry and right angled trigonometry

Key formulas

Geometry and Right Angled Trigonometry worked examples

Worked example 1: Distance and midpoint in 3D · easy

Solution

Worked example 2: Slant height and surface area of a cone · medium

Solution

Worked example 3: Angle between a 3D diagonal and a face · hard

Solution

Try these IB Maths AA SL geometry and right angled trigonometry questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 6 marks · Paper 2

Revise geometry and right angled trigonometry for the exam

FAQ

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Teaching this topic?

More AA SL Unit 3 topics