Volume & Surface Area (AI SL)
Volume and surface area let you turn a 3D shape - a cylinder, cone, sphere or pyramid - into numbers you can use: how much a tank holds, how much paint or material covers it, how much a composite solid built from several shapes weighs or costs. This topic covers the standard formulas for right-pyramids, right cones, spheres, hemispheres and cylinders, how to combine them for composite solids, and how volume and surface area scale when a solid is enlarged.
What the syllabus says
This topic maps onto the geometry and trigonometry section of the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL3.1 | Distance between two points in 3D space and their midpoint. Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane. Students should be able to identify relevant right-angled triangles within three-dimensional objects to find unknown lengths and angles. Suggested contexts: architecture and design. |
Volume and surface area content is common to both AI and AA at SL, and is examinable on both Paper 1 and Paper 2.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is volume?
Volume is the amount of three-dimensional space a solid encloses, measured in cubic units such as cm\(^3\) or m\(^3\). It answers "how much can this shape hold, or how much material does it contain?"
e.g. A cylinder of radius \(1.5\) m and height \(4\) m has volume \(\pi(1.5)^2(4)=9\pi\approx28.3\) m\(^3\).
What is surface area?
Surface area is the total area of all the outer faces or curved surfaces of a solid, measured in square units such as cm\(^2\) or m\(^2\). For a closed shape it includes every face, flat or curved.
e.g. A closed cylinder of radius \(1.5\) m and height \(4\) m has total surface area \(2\pi(1.5)^2+2\pi(1.5)(4)=16.5\pi\approx51.8\) m\(^2\).
What is a composite solid?
A composite solid is built from two or more standard shapes joined together. Its volume is the sum (or difference, if a piece is removed) of the volumes of its parts, each calculated with its own formula.
e.g. A silo made of a cylinder (radius \(2\) m, height \(6\) m) topped by a cone (radius \(2\) m, height \(1.5\) m) has total volume \(24\pi+2\pi=26\pi\approx81.7\) m\(^3\).
What is the volume scale factor for similar solids?
If two solids are similar with linear (length) scale factor \(k\), their surface areas scale by \(k^2\) and their volumes scale by \(k^3\) - volume grows much faster than length as a shape is enlarged.
e.g. Two similar cones have radii in ratio \(2:3\), so their volumes are in ratio \(2^3:3^3=8:27\); if the smaller has volume \(40\) cm\(^3\), the larger has \(40\times\tfrac{27}{8}=135\) cm\(^3\).
What is a hemisphere?
A hemisphere is exactly half a sphere, cut through its centre. Its volume is half the volume of the full sphere of the same radius: \(V=\tfrac23\pi r^3\) instead of \(\tfrac43\pi r^3\).
e.g. A hemisphere of radius \(5\) cm has volume \(\tfrac23\pi(5)^3=\tfrac{250}{3}\pi\approx262\) cm\(^3\).
Key formulas
Six formulas cover every solid on this topic. The cylinder formulas come from the booklet's prior-learning section; the cone, sphere and pyramid formulas are listed under geometry and trigonometry.
Formula reference
| Formula | Used for | Booklet? |
|---|---|---|
| \(V=\pi r^2 h\) | Volume of a cylinder | ✓ Yes - prior learning section |
| \(A=2\pi r^2+2\pi r h\) | Total surface area of a closed cylinder | ✓ Yes - prior learning section |
| \(V=\dfrac13\pi r^2 h\) | Volume of a right cone | ✓ Yes |
| \(V=\dfrac43\pi r^3\) | Volume of a sphere | ✓ Yes |
| \(A=4\pi r^2\) | Surface area of a sphere | ✓ Yes |
| \(V=\dfrac13\times\text{base area}\times\text{height}\) | Volume of a right pyramid | ✓ Yes |
Volume vs surface area
The same solid gives two very different numbers, and they scale differently when the solid is enlarged.
| Feature | Volume | Surface area |
|---|---|---|
| Measures | Space enclosed (3D) | Area of the outer faces (2D) |
| Units | cm\(^3\), m\(^3\) | cm\(^2\), m\(^2\) |
| Scaling with length ratio \(k\) | \(k^3\) | \(k^2\) |
| Example (cylinder, \(r=1.5\), \(h=4\)) | \(9\pi\approx28.3\) m\(^3\) | \(16.5\pi\approx51.8\) m\(^2\) |
Volume and surface area of standard solids
Every question on this topic uses one or more of these four solids, alone or combined.
Cylinder
\[V=\pi r^2h,\quad A=2\pi r^2+2\pi rh\]
Circular cross-section \(\times\) height, plus the curved side and two circular ends.
✓ In the formula booklet - prior learningCone
\[V=\frac13\pi r^2h\]
One third of the volume of a cylinder with the same base and height.
✓ In the formula bookletSphere & hemisphere
\[V=\frac43\pi r^3,\quad V_{\text{hemisphere}}=\frac23\pi r^3\]
A hemisphere is exactly half the volume of the full sphere of the same radius.
✓ In the formula bookletPyramid
\[V=\frac13\times\text{base area}\times\text{height}\]
One third of a prism with the same base and height - works for any base shape.
✓ In the formula bookletComposite and scaled solids
Beyond the single-shape formulas, most exam questions ask you to combine or scale them.
Composite solids
Split the solid into standard shapes, find each volume separately, then add (or subtract, if a piece is removed) - never try to force one formula over the whole thing.
Similar solids & scale factor
For a length scale factor \(k\), areas scale by \(k^2\) and volumes by \(k^3\) - keep the powers straight when scaling up or down from a known value.
Keep the exact form
Carry \(\pi\) through the whole calculation and only round at the very end - rounding an intermediate area or volume too early introduces error into the final answer.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A closed cylindrical tank has radius 1.5 m and height 4 m.
(a) Find its total surface area (3 significant figures).
(b) Paint costs $6 per m². Find the cost to paint the outside.
Worked solution
(a) Total surface area of a closed cylinder: \(A = 2\pi r^2 + 2\pi r h\) M1
with \(r=1.5,\ h=4\):
\(A = 2\pi(2.25) + 2\pi(1.5)(4) = 4.5\pi + 12\pi = 16.5\pi.\) A1
\(16.5\pi = 51.836\ldots \approx 51.8 \text{ m}^2\) (3 significant figures). A1
(b) Multiply the unrounded area by the rate $6/m²: \(51.836\ldots \times 6.\) M1
\(= 311.0\ldots\) A1
\(\approx $311\) (to the nearest dollar). A1
A sphere has surface area \(100\pi\) cm².
(a) Find its radius.
(b) Find its volume (3 significant figures).
Worked solution
(a) Solve for \(r\): \(4\pi r^2 = 100\pi.\) M1
Divide by \(4\pi\): \(r^2 = 25,\) so \(r = 5\) cm. A1
(b) Volume formula: \(V = \tfrac43 \pi r^3 = \tfrac43 \pi(5)^3.\) M1
\(=\tfrac{500}{3}\pi.\) A1
\(\tfrac{500}{3}\pi = 523.6\ldots\) A1
\(\approx 524 \text{ cm}^3\) (3 significant figures).
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Using the full sphere formula for a hemisphere. A hemisphere's volume is \(\tfrac23\pi r^3\), not \(\tfrac43\pi r^3\) - forgetting to halve it is one of the most common errors on composite solids.
- Rounding too early. Rounding \(16.5\pi\) to \(51.8\) before multiplying by a cost rate can shift the final rounded answer - keep the exact form or full calculator decimal until the last step.
- Mixing up radius and diameter. Every volume and surface area formula on this topic uses the radius \(r\) - if a question gives a diameter, halve it before substituting.
- Getting the scale-factor power wrong. Lengths scale by \(k\), areas by \(k^2\), and volumes by \(k^3\) - using the area power for a volume question (or vice versa) is a very common slip.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Volume and surface area answers usually come out as an exact multiple of \(\pi\) first - your GDC turns that into a decimal without you having to round early.
- Work out the exact form first (e.g. \(16.5\pi\) or \(26\pi\)), keeping \(\pi\) symbolic rather than rounding it to \(3.14\).
- Type the expression on the home screen and press enter; use 2nd → π for π.TI-84
- Main menu → Run-Matrix, type the expression and press EXE; π via SHIFT → EXP.Casio
- Open a Calculator page, type the expression and press enter; π via the π key.Nspire
Tip: Add the exact multiples of \(\pi\) before rounding to keep full accuracy - e.g. evaluate \(300\pi+\tfrac{250}{3}\pi\) in one go rather than rounding each part separately.
When you're given a volume or surface area and need to find a radius, height or other dimension, rearrange by hand then check the value on your GDC.
- Set the formula equal to the given value, e.g. \(4\pi r^2=100\pi\), and simplify algebraically where possible (here the \(\pi\) cancels, leaving \(r^2=25\)).
- Once solved, substitute the value back into the next formula and evaluate on the home screen.TI-84
- Use Run-Matrix to check the rearranged equation, or EQUA → Solver for a full numerical solve.Casio
- Use the Calculator page's solve() command to check the rearranged equation numerically.Nspire
Tip: The \(\pi\) often cancels when solving for a dimension - keep it symbolic until you see whether it cancels, rather than substituting a decimal approximation too early.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Volume & surface area questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between volume and surface area?
Volume measures the space enclosed inside a solid, in cubic units like \(\text{cm}^3\) or \(\text{m}^3\). Surface area measures the total area of all the outer faces or curved surfaces, in square units like \(\text{cm}^2\) or \(\text{m}^2\). A closed cylinder, for instance, has one volume figure but its surface area is the sum of two circular ends plus the curved side.
How do I find the volume of a composite solid?
Split the solid into the standard shapes it's built from (cylinder, cone, sphere, pyramid), find each volume using its own formula, then add them together - or subtract, if one shape has been removed from another, like a hole drilled through a block.
Why should I keep pi in my answer until the last step?
Rounding an intermediate value like \(16.5\pi\) to \(51.8\) before multiplying by a cost rate introduces a small error that can shift your final rounded answer. Keep the exact form (a multiple of \(\pi\), or the unrounded decimal) all the way through the calculation, and only round the very last answer.
How do I use my GDC to evaluate expressions with pi?
Type the expression directly on the home screen using the calculator's dedicated \(\pi\) key (2nd → π on a TI-84, SHIFT → EXP on a Casio, or the π key on an Nspire), then press enter or EXE. This keeps full accuracy through the calculation instead of typing in a rounded decimal. See the GDC guide for model-specific instructions.
Sub-topics
Volume & Surface Area broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AI SL syllabus unit, in case you want to keep going.