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 surface area on both Foundation and Higher papers, with curved surface area questions reserved for Higher tier. This guide covers every surface area formula you need, provides worked examples at both levels, highlights common errors and gives you practice questions. For a full list of formulas to learn, see our GCSE Maths formulas guide.

What Is Surface Area?

Surface area is the total area of all the outer faces of a three-dimensional shape. It is measured in square units (cm², m²).

Key Surface Area Formulas

Cuboid: $$SA = 2(lw + lh + wh)$$

Cube: $$SA = 6s^2$$

Prism: $$SA = 2 \times \text{cross-sectional area} + \text{sum of rectangular faces}$$

Cylinder:

• Curved surface area: $CSA = 2\pi rh$
• Total surface area: $SA = 2\pi rh + 2\pi r^2$

Cone:

• Curved surface area: $CSA = \pi rl$, where l is the slant height
• Total surface area: $SA = \pi rl + \pi r^2$

Sphere: $$SA = 4\pi r^2$$

Hemisphere: $$SA = 3\pi r^2$$ (curved surface + flat circle)

Which Formulas Are Given?

On AQA and Edexcel formula sheets, the curved surface area of a cone (πrl) and the surface area of a sphere (4πr²) are provided. Cuboid, cylinder and prism formulas are not given — memorise these.

Step-by-Step Method

1. Identify the 3D shape and all its faces.
2. List each face and its shape (rectangle, triangle, circle, curved surface).
3. Calculate the area of each face using the appropriate 2D formula.
4. Add all face areas together to get the total surface area.
5. Include square units in your answer.

Nets

Imagining a shape unfolded into its net can help. A cylinder unfolds into two circles and a rectangle. A cone unfolds into a circle and a sector.

Worked Example 1 — Foundation Level

A cuboid measures 8 cm by 5 cm by 3 cm. Find its total surface area.

Step 1: Identify the three pairs of faces:

• Top and bottom: 8 × 5 = 40 cm² each → 80 cm²
• Front and back: 8 × 3 = 24 cm² each → 48 cm²
• Left and right: 5 × 3 = 15 cm² each → 30 cm²

Step 2: Total = 80 + 48 + 30 = 158 cm².

Or using the formula: SA = 2(40 + 24 + 15) = 2 × 79 = 158 cm².

Worked Example 2 — Higher Level

A solid is made from a cone placed on top of a cylinder. The cylinder has radius 4 cm and height 10 cm. The cone has the same radius and a slant height of 7 cm. Find the total surface area of the solid. Give your answer to 1 decimal place.

Step 1: The base of the solid is the bottom circle of the cylinder. $$A_{\text{base}} = \pi \times 4^2 = 16\pi$$

Step 2: Curved surface of the cylinder. $$CSA_{\text{cyl}} = 2\pi \times 4 \times 10 = 80\pi$$

Step 3: Curved surface of the cone. (The cone sits on top of the cylinder, so its base circle is not an outer face.) $$CSA_{\text{cone}} = \pi \times 4 \times 7 = 28\pi$$

Step 4: Total surface area = base + curved cylinder + curved cone. $$SA = 16\pi + 80\pi + 28\pi = 124\pi = 389.557\ldots$$

Answer: 389.6 cm² (1 d.p.).

Note: we did not include the circle where the cone meets the cylinder — that is an internal face.

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.
• Not pairing faces for cuboids. A cuboid has three different rectangular faces, each appearing twice — multiply by 2.

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.

Practice Questions

Question 1 (Foundation) A cube has side length 4 cm. Find its surface area.

Question 2 (Foundation) A cylinder has radius 3 cm and height 10 cm. Find its total surface area to 1 decimal place.

Question 3 (Higher) A cone has radius 5 cm and perpendicular height 12 cm. Find the total surface area. Give your answer to 3 significant figures.

Question 4 (Higher) A sphere has a surface area of 400π cm². Find its radius.

Question 5 (Higher) A hemisphere of radius 6 cm sits on top of a cylinder of radius 6 cm and height 8 cm. Find the total surface area of the solid to 1 decimal place.

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Related Topics

• Volume of 3D Shapes — volume and surface area are often tested together
• Area of 2D Shapes — the building blocks for surface area calculations
• Pythagoras' Theorem — needed to find slant heights
• Nets of 3D Shapes — visualising the unfolded shape aids understanding
• Unit Conversions — converting cm² to m² when required

Summary

Surface area is the total area of all outer faces of a 3D shape. For cuboids, add the areas of all six faces. For cylinders, add the curved surface area (2πrh) to the area of the circles. For cones, use πrl for the curved surface and add πr² for the base. For spheres, use 4πr². Always check whether the shape is open or closed, whether faces are internal (hidden), and whether you need the slant height or perpendicular height. Show each face area separately in your working, include square units, and check your answer is sensible.