Sheet № 47 · Foundation + Higher · AQA · Edexcel · OCR
Surface Area –
Surface area questions test your ability to visualise three-dimensional shapes and apply two-dimensional area formulas to each face — a skill that examiners consistently assess at GCSE. From painting a room (cuboid) to wrapping a cylindrical candle, these questions connect geometry to everyday life. AQA, Edexcel and OCR all include surfac
§Key definitions
Step 1:
Identify the three pairs of faces:
Step 2:
Total = 80 + 48 + 30 = 158 cm².
Step 3:
Curved surface of the cone. (The cone sits on top of the cylinder, so its base circle is not an outer face.)
Step 4:
Total surface area = base + curved cylinder + curved cone.
Answer: 389.6 cm²
(1 d.p.).
§Formulas to memorise
SA = 2(lw + lh + wh)
SA = 6s^2
SA = 2 \times \text{cross-sectional area} + \text{sum of rectangular faces}
SA = 4\pi r^2
SA = 3\pi r^2
A_{\text{base}} = \pi \times 4^2 = 16\pi
CSA_{\text{cyl}} = 2\pi \times 4 \times 10 = 80\pi
CSA_{\text{cone}} = \pi \times 4 \times 7 = 28\pi
SA = 16\pi + 80\pi + 28\pi = 124\pi = 389.557\ldots
Worked example
See example below.
A cuboid measures 8 cm by 5 cm by 3 cm. Find its total surface area.
⚠ Common mistakes
- ✗Including internal faces. When two shapes are joined (e.g. hemisphere on a cylinder), the face where they meet is hidden. Do not count it.
- ✗Confusing slant height and perpendicular height. The cone CSA formula uses the slant height l, not the vertical height h. If only h is given, use Pythagoras: l² = r² + h².
- ✗Forgetting the base. A closed cylinder has two circles; an open-topped cylinder has one. Read the question carefully.
- ✗Using diameter instead of radius. All formulas use r.
- ✗Mixing up surface area and volume formulas. Surface area is in cm²; volume is in cm³. If your answer has cubic units, you have used the wrong formula.
✦ Exam tips
- →Sketch the net if you are unsure which faces to include. This makes it visual and prevents missed faces.
- →Write each face area separately before adding them. This earns method marks and helps you keep track.
- →For cylinders, remember the curved surface unrolls to a rectangle with width = circumference (2πr) and height = h. This can help you remember the formula.
- →Pythagoras and surface area are often combined. If you need the slant height of a cone or the length of a slanting edge, use a² + b² = c².
- →Leave in terms of π if instructed, otherwise use the calculator's π button for accuracy.