Geometry · GCSE Maths
Surface area
GCSE Maths surface area: cuboids, cylinders (2πr² + 2πrh), spheres (4πr²), and nets where they help you see every face before adding areas.
Surface area is the sum of all outer faces. Unwrap the solid into a net, find each face area, add — do not confuse with volume.
The important bits
What you need to know
- 1
Cuboid: SA = 2(lw + lh + wh). There are three distinct face pairs; each pair appears twice on the surface.
- 2
Cube: SA = 6a², since all six faces are squares of side a.
- 3
Cylinder (closed): SA = 2πr² + 2πrh. Two circular ends (2πr²) plus curved surface (2πrh). The curved part unwraps to a rectangle of width 2πr and height h.
- 4
Sphere: SA = 4πr². This is not the same as volume 4/3 πr³ — squared versus cubed radius.
- 5
Nets: drawing the net helps you list every face. A cuboid net has six rectangles; label matching pairs before adding.
- 6
Open solids: if the top is missing (open box, no lid), subtract that face from the full cuboid surface area.
- 7
Units are square: cm², m². Do not write cm³ for surface area.
- 8
Show the breakdown: “two circles: … + curved: … = total” scores method marks even if arithmetic slips later.
Quotations worth analysing
Short evidence. Real method.
“SA = 2πr² + 2πrh”
2πr² is both circular ends. 2πrh is the curved surface — the circumference 2πr times the height h. Missing one circle is the usual open-ended cylinder error.
“SA = 4πr²”
No edges on a sphere — one continuous curved surface. Do not add “extra faces” or use 4/3 πr³, which is volume.
“2(lw + lh + wh)”
The bracket is three different face areas. Doubling accounts for top and bottom, front and back, left and right.
Go deeper
Cuboid nets and missing faces
A cuboid 3 cm × 4 cm × 5 cm has faces 12, 15 and 20 cm². SA = 2(12 + 15 + 20) = 94 cm². Alternatively list: two of 12, two of 15, two of 20. An open box (no lid) with the same dimensions: subtract one 3 × 4 face → 94 − 12 = 82 cm². Read the question: “open at the top” means one face is not painted. Cylinder open at one end: one circle plus curved only → πr² + 2πrh. Always name which faces you include. A net diagram in your working is strong evidence of method.
Go deeper
Why the cylinder has 2πrh
Cut the curved surface vertically and flatten it: you get a rectangle. Width = distance around the circle = 2πr. Height = h. Area = 2πr × h = 2πrh. Add the two circular ends, each πr². Example: r = 4 cm, h = 10 cm. Circles: 2 × π × 16 = 32π cm². Curved: 2 × π × 4 × 10 = 80π cm². Total 112π cm² ≈ 352 cm² to 3 s.f. If only the curved surface is needed (label on a can), use 2πrh alone.
Go deeper
Surface area versus volume in exam questions
Volume fills the inside (cm³, litres). Surface area covers the outside (cm²), e.g. paint or wrapping paper. A question giving radius and height might ask for both — use πr²h for volume and 2πr² + 2πrh for full surface area. Sphere SA = 4πr²; sphere V = 4/3 πr³. Write which formula you are using at the top of your working. Compound solids: find the surface area of each exposed part; do not add internal faces where two parts join. A hemisphere on a cylinder: include the curved hemisphere, the cylinder curved face, one circle base, but not the circle where they glue together if that face is hidden.
See the idea in action
Find the surface area of a closed cylinder with radius 5 cm and height 8 cm, in terms of π. Step 1: SA = 2πr² + 2πrh. Step 2: Two circles: 2 × π × 5² = 2 × 25π = 50π cm². Step 3: Curved surface: 2 × π × 5 × 8 = 80π cm². Step 4: Total SA = 50π + 80π = 130π cm². Check: curved part is larger than the two ends combined here because h > r.
Exam technique
Turn knowledge into marks
Sketch a net or list faces before calculating. For cylinders, write “2 circles” and “curved rectangle” on separate lines, then add.
Common mistakes
Do not give these marks away
- 01
Using 2πrh alone for a closed cylinder and forgetting the two circular ends.
- 02
Confusing surface area 4πr² with volume 4/3 πr³ for a sphere.
- 03
Including a face that is internal or excluded (e.g. adding the lid on an open box).
A cube has side length 3 cm. What is its surface area?
A54 cm²
B27 cm²
C18 cm²
D9 cm²
Show the answer
54 cm². SA = 6a² = 6 × 9 = 54 cm². 27 cm³ is the volume. 18 cm² might be three faces only. 9 cm² is one face.
Quick questions
If this is the bit you searched
What is the surface area formula for a cylinder?
Closed cylinder: 2πr² + 2πrh. Open at one end: πr² + 2πrh. Curved surface only: 2πrh.
How does a net help?
It shows every face flat. Label each rectangle or circle, find its area, then add. Useful for cuboids and prisms.
Is sphere surface area the same as volume?
No. Surface area is 4πr² (square units). Volume is 4/3 πr³ (cubic units).
Why is cuboid SA doubled?
Each of the three face types appears twice on opposite sides of the cuboid, so we calculate lw + lh + wh once and multiply by 2.