Surface Area (AI SL)
Surface area asks a different question from volume: not how much a solid holds, but how much material would cover the outside of it. That means adding up flat and curved faces separately, and being precise about which faces actually exist - an open-topped box has one fewer face than a closed one, and a frustum has two circular ends of different sizes rather than one. It's part of the broader Volume & Surface Area topic.
24 questions on this sub-topic.
Two key formulas
Covered under IB syllabus reference SL3.1: volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. Both formulas below are given in the formula booklet.
Closed cylinder
\(A=2\pi r^2+2\pi r h\)
Two circular ends (\(2\pi r^2\)) plus the curved side unrolled into a rectangle (\(2\pi r h\)).
Sphere
\(A=4\pi r^2\)
A hemisphere needs the curved half of this, \(2\pi r^2\), plus its flat circular face if the question asks for total surface area.
Need the full syllabus wording and formula-booklet reference table? See Volume & Surface Area, or the calculator steps at using your GDC.
Worked examples
Find the surface area of a sphere of radius 7 cm, giving your answer as a multiple of \(\pi\) and to 3 significant figures.
Worked solution
Surface area of a sphere: \(A = 4\pi r^2\). M1
\(r=7\): \(A = 4\pi (7)^2 = 4\pi(49) = 196\pi.\) A1
\(196\pi = 615.75\ldots \approx 616 \text{ cm}^2\) (3 significant figures). A1
A frustum is formed by cutting a small cone (radius 2 cm, slant height 3 cm) from a large cone (radius 5 cm, slant height 12 cm).
Find the total surface area of the frustum (3 significant figures).
Worked solution
Curved area of frustum \(= \pi R L - \pi r l\):
\(\pi(5)(12) - \pi(2)(3) = 60\pi - 6\pi\) M1 \(= 54\pi.\) A1
Large base: \(\pi(5)^2 = 25\pi\). Small top: \(\pi(2)^2 = 4\pi\). M1
\(54\pi + 25\pi + 4\pi = 83\pi = 260.7\ldots\) A1 \(\approx 261 \text{ cm}^2\) (3 significant figures). A1
Common mistakes
- Mixing up radius and diameter. Every 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 wrong power when a solid is enlarged is a very common slip on surface-area questions.
- Including a face that isn't really there. An open-topped container is missing a face, and a frustum has two different-sized circular ends rather than one - always sketch which flat faces genuinely exist before adding them up.
Ready to practise properly?
24 surface-area questions, marked instantly like the real exam.
Quick answers
What is the formula for the surface area of a sphere?
The surface area of a sphere of radius \(r\) is \(A = 4\pi r^2\). This is in the formula booklet.
What is the formula for the total surface area of a closed cylinder?
The total surface area of a closed cylinder of radius \(r\) and height \(h\) is \(A = 2\pi r^2 + 2\pi r h\), made up of the two circular ends and the curved side.