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