Knowledge Organiser: Volume & Surface Area of Spheres
Key Terms
- Sphere: A perfectly round 3D shape: every point on the surface is the same distance from the centre
- Hemisphere: Exactly half a sphere
- Radius (r): Distance from centre to surface
Must-Know Facts
- Both sphere formulas are given in the exam question when needed (they are not on the formula sheet): know how to use them
- Hemisphere volume = ½ × (4/3)πr³ = (2/3)πr³
- Hemisphere total surface area = curved half + flat circle = 2πr² + πr² = 3πr²
- Volume grows with r³: doubling radius multiplies volume by 8
- Surface area grows with r²: doubling radius multiplies SA by 4
Key Formulas
- Sphere volume: V = (4/3)πr³
- Sphere surface area: SA = 4πr²
- Hemisphere volume: V = (2/3)πr³
- Hemisphere total SA: SA = 3πr²
Common Mistakes
- Using diameter instead of radius: Both sphere formulas use r: if given d, halve it first
- Hemisphere surface area: Total SA = 2πr² (curved) + πr² (flat circle) = 3πr²: don't forget the flat face
- Volume formula: V = (4/3)πr³: the power is 3, not 2
- Exact answers: Leave as e.g. 36π cm³ when asked for exact answers in terms of π
Worked Examples
Find the volume of a sphere with radius 6cm. Solution V = (4/3)πr³ V = (4/3) × π × 6³ V = (4/3) × π × 216 = 288π = 904.8 cm³ (1 d.p.)