IB Maths AI SL · Unit 1: Number and Algebra
IB Maths AI SL Volume and Surface Area Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 36 questions
- Paper 1: 21
- Paper 2: 15
- 6 easy
- 13 medium
- 10 hard
- 3 very hard
- 4 starter
- 5 worked examples
Practise Volume and Surface Area questions →
AI SL formula booklet
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 (9)
- 3D Geometric Measures (8)
- Basic 3D Shapes (6)
- Volume and Area Problems (5)
- Composite Solids (4)
- Composite & Related Solids (4)
Volume and Surface Area worked examples
Worked example 1: Sphere volume in terms of π and surface area · easy
A spherical globe has radius $4.5\text{ cm}$. 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: $254.469\ldots \implies \mathbf{254\text{ cm}^2}$.
Examiner tip: "Exact in terms of $\pi$" means LEAVE the $\pi$ 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 $5\text{ cm}$ 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: $282.74\ldots \implies \mathbf{283\text{ cm}^2}$.
Examiner tip: The formula booklet gives $\pi r l$ (curved surface only). For a SOLID cone you must also add the circular base $\pi r^2$ — 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 $3\text{ cm}$. 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: $V = 90\pi + 18\pi = 108\pi \implies 339.292\ldots \implies \mathbf{339\text{ cm}^3}$.
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 $6\text{ m}$. Find the volume to 3 s.f.
1. Formula: $V = \pi r^2 h = \pi(6.25)(6) = 37.5\pi$.
2. State: $\mathbf{118\text{ m}^3}$.
Examiner tip: Convert all dimensions to the same unit BEFORE substituting.
Worked example 5: Closed cylinder surface area · medium
Solid cylinder: diameter $8$, length $15\text{ cm}$. 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: $\mathbf{478\text{ cm}^2}$.
Examiner tip: "Solid" = curved surface + two flat ends. "Open pipe" = just $2\pi r h$.
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.
Question 1 · easy · 5 marks · Paper 1
A piece of candy is made in the shape of a solid hemisphere. The radius of the hemisphere is \(6\text{ mm}\).
Calculate the total surface area of one piece of candy.
Calculate the volume of one piece of candy.
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 $r$.
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 $k\pi$, where $k$ is an integer.
All 36 volume and surface area questions with mark schemes →
FAQ
How many IB Maths AI SL volume and surface area questions are there?
There are 36 exam-style volume and surface area questions in the AI SL question bank (Paper 1: 21 · Paper 2: 15), graded 6 easy, 13 medium, 10 hard, 3 very hard, 4 starter. 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: 21 · 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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