A spherical globe has radius $4.5\text{ cm}$. Calculate the exact volume in terms of $\pi$, and the surface area to 3 s.f.

1. Recall : $V = \frac{4}{3}\pi r^3$. 2. Substitute : $V = \frac{4}{3}\pi (4.5)^3 = \frac{4}{3}\pi (91.125)$. 3. Evaluate exactly: $V = \mathbf{121.5\pi\text{ cm}^3}$. 4. Recall : $SA = 4\pi r^2$. 5. Substitute : $SA = 4\pi (4.5)^2 = 81\pi$. 6. Evaluate : $254.469\ldots \implies \mathbf{254\text{ cm}^2}$.

A solid right circular cone has base radius $5\text{ cm}$ and vertical height $12\text{ cm}$. Calculate the total surface area of the cone.

1. Recognise : total surface area $=$ curved surface ($\pi r l$) $+$ circular base ($\pi r^2$). 2. Slant height by Pythagoras: $l = \sqrt{5^2 + 12^2} = \sqrt{169} = 13\text{ cm}$. 3. Base area : $A_{base} = \pi(5)^2 = 25\pi$. 4. Curved surface : $CSA = \pi(5)(13) = 65\pi$. 5. Sum : total $SA = 25\pi + 65\pi = 90\pi$. 6. Evaluate : $282.74\ldots \implies \mathbf{283\text{ cm}^2}$.

A wooden toy is a solid hemisphere on top of a solid cylinder. Both have radius $3\text{ cm}$. The overall vertical height of the toy is $13\text{ cm}$. Calculate the total volume of the toy.

1. Determine the cylinder height: hemisphere adds $r = 3\text{ cm}$ on top, so cylinder $h = 13 - 3 = 10\text{ cm}$. 2. Recall : $V_{cyl} = \pi r^2 h$. 3. Cylinder : $V_{cyl} = \pi(3)^2(10) = 90\pi\text{ cm}^3$. 4. Recall : $V_{hemi} = \frac{1}{2} \cdot \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$. 5. Hemisphere : $V_{hemi} = \frac{2}{3}\pi(27) = 18\pi\text{ cm}^3$. 6. Sum : $V = 90\pi + 18\pi = 108\pi \implies 339.292\ldots \implies \mathbf{339\text{ cm}^3}$.

Cylindrical tank: radius $2.5$, height $6\text{ m}$. Find the volume to 3 s.f.

1. Formula : $V = \pi r^2 h = \pi(6.25)(6) = 37.5\pi$. 2. State : $\mathbf{118\text{ m}^3}$.

Solid cylinder: diameter $8$, length $15\text{ cm}$. Find the total surface area.

1. Radius : $r = 4$. 2. SA : $2\pi rh + 2\pi r^2 = 120\pi + 32\pi = 152\pi$. 3. State : $\mathbf{478\text{ cm}^2}$.

Home › IB Maths AI SL › Questions by topic › Volume and Surface Area

IB Maths AI SL · Unit 1: Number and Algebra

IB Maths AI SL Volume and Surface Area Questions

Updated October 2026

Exam-style IB Maths AI SL volume and surface area questions with worked solutions. Start with the 5 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.

35 questions
Paper 1: 20
Paper 2: 15
7 easy
15 medium
10 hard
3 very hard
5 worked examples

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What's examined in AI SL volume and surface area

The question bank covers these volume and surface area question types (number of questions in brackets):

Direct V/SA Application (8)
3D Geometric Measures (8)
Basic 3D Shapes (6)
Volume and Area Problems (5)
Composite Solids (4)
Composite & Related Solids (4)

Where students lose marks on volume and surface area

Examiner's noteTriangles and 3D: wrong rule, wrong angle, missing units

Where marks are lost: Candidates assumed a right angle that was not there (or used the sine rule in a right-angled triangle), found an angle other than the one named, used the wrong radius or only one of two areas, missed an arc length in a curved surface, forgot units where they were required, and did not give an answer in standard form when asked. Drawing a diagram helped.

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

Volume and Surface Area worked examples

Worked example 1: Sphere volume in terms of π and surface area · easy

A spherical globe has radius $4.5 cm$ . Calculate the exact volume in terms of $π$ , and the surface area to 3 s.f.

Solution

6 steps

Examiner tip: "Exact in terms of

π

" means LEAVE the

π

symbol in your final answer — do not evaluate it as a decimal.

Worked example 2: Total surface area of a cone (Pythagoras for slant height) · medium

A solid right circular cone has base radius $5 cm$ and vertical height $12 cm$ . Calculate the total surface area of the cone.

Solution

6 steps · 1 to fill in

Examiner tip: The formula booklet gives

π r l

(curved surface only). For a SOLID cone you must also add the circular base

π r^{2}

— the booklet won't do it for you.

Worked example 3: Composite volume — cylinder plus hemisphere · hard

A wooden toy is a solid hemisphere on top of a solid cylinder. Both have radius $3 cm$ . The overall vertical height of the toy is $13 cm$ . Calculate the total volume of the toy.

Solution

6 steps

Examiner tip: In composite 3-D solids, subtract the radius of any hemispherical or conical cap from the overall vertical height to find the true height of the base cylinder / prism.

Worked example 4: Cylinder volume · easy

Cylindrical tank: radius $2.5$ , height $6 m$ . Find the volume to 3 s.f.

Solution

2 steps

Examiner tip: Convert all dimensions to the same unit BEFORE substituting.

Worked example 5: Closed cylinder surface area · medium

Solid cylinder: diameter $8$ , length $15 cm$ . Find the total surface area.

Solution

3 steps

Examiner tip: "Solid" = curved surface + two flat ends. "Open pipe" = just

2 π r h

Try these IB Maths AI SL volume and surface area 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 solid right square-based pyramid has a base side length of $10 m$ and a vertical height of $12 m$ . Calculate the angle between one of the sloping triangular faces and the square base.

Attempt it and see the mark scheme →

Question 2 · medium · 4 marks · Paper 2

A cylinder has a radius of $r$ cm and a height of $12$ cm. Its volume is equal to the volume of a sphere with a radius of $6$ cm. Find the exact value of $r$ .

Full worked solution Free, no sign-in

Volume of the sphere: $V = \frac{4}{3} π (6^{3}) = 288 π$ M1
Equate to the cylinder volume: $π r^{2} (12) = 288 π$ M1
Divide by $12 π$ : $r^{2} = 24$ A1
$r = \sqrt{24} = 2 \sqrt{6}$ cm A1

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

Question 3 · hard · 4 marks · Paper 2

A solid wooden cylinder has a radius of $5$ cm and a height of $12$ cm. Calculate its total surface area, giving your answer in the exact form $k π$ , where $k$ is an integer.

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI SL volume and surface area questions are there?

There are 35 exam-style volume and surface area questions in the AI SL question bank (Paper 1: 20 · Paper 2: 15), graded 7 easy, 15 medium, 10 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is volume and surface area on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 20 · Paper 2: 15. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 1 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): Volume & surface area of 3D shapes.

IB Maths AI SL Volume and Surface Area Questions

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What's examined in AI SL volume and surface area

Where students lose marks on volume and surface area

Volume and Surface Area worked examples

Worked example 1: Sphere volume in terms of π and surface area · easy

Solution

Worked example 2: Total surface area of a cone (Pythagoras for slant height) · medium

Solution

Worked example 3: Composite volume — cylinder plus hemisphere · hard

Solution

Worked example 4: Cylinder volume · easy

Solution

Worked example 5: Closed cylinder surface area · medium

Solution

Try these IB Maths AI SL volume and surface area questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 4 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 4 marks · Paper 2

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