GCSE Maths Volume & Surface Area Notes
Volume measures the space inside a 3D solid; surface area is the total area of all its faces. Prisms use the cross-section, and cylinders and spheres use π.
Key concepts
- Volume of a prism
- Area of the cross-section × length — works for any prism, including cuboids.
- Cylinder
- A circular prism: volume = πr²h; curved surface area = 2πrh.
- Surface area
- The sum of the areas of every face of the solid.
- Sphere
- Volume = ⁴⁄₃πr³ and surface area = 4πr² (given on the formula sheet).
Key formulas
Prism volume
V = cross-section area × length
Cylinder volume
V = πr²h
Cuboid surface area
SA = 2(lw + lh + wh)
Worked example
Find the volume of a cylinder with radius 3 cm and height 10 cm (to 1 d.p.).
- 1Use V = πr²h.
- 2Substitute: V = π × 3² × 10 = π × 9 × 10 = 90π.
- 3Work it out: 90 × 3.14159… = 282.743…
- 4Round to 1 decimal place: 282.7 cm³.
Answer: 282.7 cm³
Common mistakes
✗ Confusing volume with surface area.
✓ Volume fills the inside (cm³); surface area covers the outside (cm²).
✗ Using diameter instead of radius for a cylinder.
✓ Halve the diameter to get r before using πr²h.
✗ Forgetting a face when finding surface area.
✓ Count every face — a cuboid has six.
Quick check
Try these, then reveal the answers.
Volume of a cuboid 2 × 3 × 4?Reveal answer
24 cm³
Volume of a prism, cross-section 12 cm², length 5 cm?Reveal answer
60 cm³
Units for surface area?Reveal answer
Square units, e.g. cm².
Exam tips
- For any prism, find the cross-section area first, then multiply by the length.
- Sketch the net to make sure you include every face for surface area.
- Give volume in cubic units and surface area in square units.
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Frequently asked questions
How do I find the volume of a prism?+
Multiply the area of its constant cross-section by its length.
What is the volume of a cylinder?+
V = πr²h — the area of the circular end times the height.
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