Surface Area & Volume
Volume vs Surface Area
Volume measures how much space a 3D shape occupies — think of filling it with water. Units: cubic units (cm³, m³, mm³)
Surface area measures the total area of all outer faces — think of wrapping paper needed to cover it. Units: square units (cm², m², mm²)
Always check your units — mixing cm² and cm³ is one of the most common errors in this topic.
Volume Formulas
| Shape | Formula |
|---|---|
| Cuboid | l × w × h |
| Cube | s³ |
| Prism (any) | Area of cross-section × length |
| Cylinder | πr²h |
| Cone | (1/3)πr²h |
| Pyramid | (1/3) × base area × height |
| Sphere | (4/3)πr³ |
Surface Area Formulas
| Shape | Formula |
|---|---|
| Cuboid | 2(lw + lh + wh) |
| Cube | 6s² |
| Cylinder | 2πr² + 2πrh |
| Cone | πr² + πrl (l = slant height) |
| Sphere | 4πr² |
Cuboid: Worked Examples
Cuboid 5cm × 4cm × 3cm:
Volume = 5 × 4 × 3 = 60cm³
Surface area: 6 faces in 3 pairs
- 2 faces of 5 × 4 = 40
- 2 faces of 5 × 3 = 30
- 2 faces of 4 × 3 = 24
SA = 40 + 30 + 24 = 94cm²
Or use the formula: 2(lw + lh + wh) = 2(20 + 15 + 12) = 2 × 47 = 94cm²
Cube with side 4cm:
Volume = 4³ = 64cm³
SA = 6 × 4² = 6 × 16 = 96cm²
Cylinder: Worked Example
A cylinder has two circular ends and a curved surface that unrolls into a rectangle.
Cylinder: radius 3cm, height 10cm (π ≈ 3.14)
Volume = πr²h = 3.14 × 9 × 10 = 282.6cm³
Surface area = 2πr² + 2πrh = 2 × 3.14 × 9 + 2 × 3.14 × 3 × 10 = 56.52 + 188.4 = 244.92cm²
For the surface area of radius 2cm, height 5cm: SA = 2π(4) + 2π(2)(5) = 8π + 20π = 28π cm² ≈ 87.96cm²
Prism: Worked Example
A prism has the same cross-section all the way along its length.
Volume = Area of cross-section × length
Triangular prism: triangle area = 12cm², length = 8cm
Volume = 12 × 8 = 96cm³
Surface area = 2 × (triangle area) + 3 rectangular faces (each = side length × prism length)
Cone and Pyramid
Both are one-third the volume of the corresponding cylinder or prism with the same base and height.
Cone: radius 3cm, height 4cm
Volume = (1/3) × π × 3² × 4 = (1/3) × 36π = 12π cm³ ≈ 37.7cm³
For cone surface area, the slant height l is needed: l = √(r² + h²) = √(9 + 16) = √25 = 5cm
SA = πr² + πrl = π(9) + π(3)(5) = 9π + 15π = 24π cm²
Sphere: Worked Example
Sphere: radius 3cm (π ≈ 3.14)
Volume = (4/3) × π × 3³ = (4/3) × π × 27 = 36π ≈ 113.1cm³
Surface area = 4π × 3² = 36π ≈ 113.1cm²
(Note: for a sphere of radius 3, volume and surface area happen to give the same numerical value — this is a coincidence for r = 3.)
Scale Factor and Volume
When a shape is enlarged by a linear scale factor k:
- Lengths multiply by k
- Areas multiply by k²
- Volumes multiply by k³
Example: A cube of side 2cm has volume 8cm³. Scale factor 3 → new volume = 8 × 3³ = 8 × 27 = 216cm³ (side = 6cm, check: 6³ = 216 ✓)
Common Mistakes
- Using diameter instead of radius in formulas involving r (radius = diameter ÷ 2)
- Giving volume in cm² or surface area in cm³ — always use the correct unit
- Forgetting to include both circular ends when finding cylinder surface area
- Using the height instead of the slant height for cone surface area
- Not multiplying by (1/3) for cone and pyramid volumes
Tips and Tricks
- Learn the formula table — many exam papers provide it, but knowing it saves time
- For any prism: volume = cross-section area × length
- Cylinder = circular prism: same rule applies
- Always square the radius (r²) in circle-based formulas — a common slip is using r instead of r²
- Sketch the shape and label all measurements before substituting into a formula