📦 Volume

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

  • Volume of cuboids and prisms (V = base area × height)
  • Volume of cylinders (V = πr²h)
  • Volume of pyramids and cones (V = ⅓ × base area × height)
  • Volume of spheres (V = ⁴⁄₃πr³)
  • Solving problems with volume and capacity

📖 Learn

1. Prisms (Cuboid and General)

V = base area × height (length)
Cuboid: V = l × w × h
Triangular prism: V = (½ × b × h_triangle) × length
Trapezoid prism: V = ½(a+b)h_trap × length
Example (cuboid): 5 × 4 × 3 = 60 cm³
💡 Key: find the cross-section area first, then multiply by the length of the prism.

2. Cylinder

V = πr²h
r = radius of circular cross-section, h = height
Example: r = 4 cm, h = 10 cm → V = π × 16 × 10 = 160π ≈ 502.7 cm³
From diameter: r = d/2 first
💡 A cylinder is a circular prism: V = (πr²) × h

3. Pyramid and Cone

V = ⅓ × base area × height
Square pyramid: V = ⅓ × s² × h
Cone: V = ⅓ × πr² × h
Example (cone): r = 3, h = 7 → V = ⅓ × π × 9 × 7 = 21π ≈ 65.97 cm³
💡 Pyramid/cone = ⅓ of the matching prism/cylinder with same base and height.

4. Sphere

V = ⁴⁄₃πr³
Example: r = 6 cm → V = (4/3)π(216) = 288π ≈ 904.8 cm³
Hemisphere: V = ½ × (4/3)πr³ = (2/3)πr³
💡 Remember r³ means r × r × r. Cube the radius before multiplying!

5. Units and Capacity

ConversionMultiplier
cm³ → ml1 cm³ = 1 ml
cm³ → litres÷ 1000
m³ → litres× 1000
m³ → cm³× 1 000 000
💡 1 litre = 1000 ml = 1000 cm³. 1 m³ = 1 000 000 cm³.

✏️ Worked Examples

Example 1 – Triangular Prism

A triangular prism has cross-section: triangle base 6 cm, height 4 cm. Prism length = 10 cm. Find volume.

Base area: ½ × 6 × 4 = 12 cm²
Volume: 12 × 10 = 120 cm³

Example 2 – Cylinder

Cylinder r = 5 cm, h = 8 cm. Find volume to 1 d.p.

V = π × 25 × 8 = 200π ≈ 628.3 cm³

Example 3 – Cone

Cone r = 4 cm, h = 9 cm. Find volume to 2 d.p.

V = ⅓ × π × 16 × 9 = 48π ≈ 150.80 cm³

Example 4 – Sphere to Capacity

A spherical tank has radius 30 cm. Find volume in litres (to 1 d.p.).

V = (4/3)π(30³) = (4/3)π(27000) = 36000π cm³
In litres: 36000π ÷ 1000 ≈ 113097 ÷ 1000 ≈ 113.1 litres

🎨 Visualizer

📦 Volume Calculator

💧 Capacity Converter

Exercise 1 – Cuboid Volume

V = l × w × h

Exercise 2 – Prism Volume (V = base area × length)

Find the cross-section area first, then multiply by the prism length.

Exercise 3 – Cylinder Volume

V = πr²h. Give to 2 d.p.

Exercise 4 – Cones, Pyramids and Spheres

To 2 d.p.

Exercise 5 – Missing Dimensions and Capacity

📝 Practice – 20 Questions

🏆 Challenge – 8 Questions