Volume & Surface Area (AA SL)
Most volume and surface area questions are really about picking the right formula from the booklet and substituting carefully - the geometry itself is rarely the hard part. This page covers the pyramid, cone and sphere formulas, plus the compound-solid traps that catch students under time pressure. It's part of the broader 3D Geometry topic.
15 questions on this sub-topic.
Key formulas
Covered under IB syllabus reference SL3.1: volume and surface area of three-dimensional solids including the right-pyramid, right cone, sphere, hemisphere, and combinations of these solids. All four formulas below are given in the formula booklet.
Volumes
\(V=\dfrac13\times\text{base area}\times\text{height}\)
\(V=\dfrac13\pi r^2 h\)
\(V=\dfrac43\pi r^3\)
Pyramid, cone, and sphere respectively. A cylinder's volume, \(V=\pi r^2 h\), isn't in the booklet but follows the same "base area times height" logic as a prism.
Surface area
\(A=4\pi r^2\)
Surface area of a sphere - the one booklet result you can't derive quickly, so it's worth just knowing where to find it. Cone, pyramid and cylinder surface areas are built from the areas of their individual flat and curved faces added together.
Need the full syllabus wording and the 3D trigonometry that often shares a question with this? See 3D Geometry.
Worked examples
Find the volume of a cone with radius 5 cm and height 12 cm (3 significant figures).
Worked solution
The volume of a cone is \(V = \tfrac13 \pi r^2 h\). M1
Substituting \(r=5,\ h=12\): \(V = \tfrac13 \pi (5)^2 (12) = 100\pi = 314\ \text{cm}^3\) (3 significant figures). A1
A cylinder has volume 500 cm³ and radius 4 cm.
Find its height (3 significant figures).
Worked solution
A cylinder's volume is \(V = \pi r^2 h\). Substituting \(V=500,\ r=4\): \(500 = \pi(4)^2 h = 16\pi h.\) M1 A1
Rearranging, \(h = \dfrac{500}{16\pi} = 9.95\ \text{cm}\) (3 significant figures). M1 A1
Common mistakes
- Using diameter instead of radius. Every booklet formula for a cone, sphere or hemisphere uses radius \(r\) - if the question gives a diameter, halve it before substituting anything.
- Double-counting or missing joined faces in compound solids. When two solids are glued together (a cone sat on a cylinder, say), the surface where they meet is internal and must be left out of the total surface area, even though both full volumes still count.
- Substituting before finding a missing length. A cone's surface area needs its slant height, not its vertical height - if only the vertical height is given, find the slant height with Pythagoras first rather than plugging the wrong length straight into the formula.
Ready to practise properly?
15 volume and surface area questions, marked instantly like the real exam.
Quick answers
Do I need to memorise the volume and surface area formulas?
No. Volume of a pyramid, cone and sphere, and surface area of a sphere, are all given in the formula booklet under SL3.1. You do need to know which formula fits which solid.
How do I find the surface area of a compound solid?
Add the areas of the visible outer faces only. Any face where two solids join is internal and must be left out of the total, even though it still counts for volume.