Sphere Formulas
A sphere is a perfectly round three-dimensional shape where every point on its surface is equidistant from its center. This distance is the radius (r). Common examples: Earth (approximately), billiard balls, bubbles, planets, and ball bearings.
Surface Area SA = 4 × π × r² ≈ 12.566 × r²
Diameter d = 2r
From diameter: r = d/2
Why is Volume (4/3)πr³?
Archimedes proved around 250 BC that a sphere's volume equals exactly 2/3 of the volume of the smallest cylinder that can contain it. That cylinder has radius r and height 2r, giving volume = π × r² × 2r = 2πr³. Therefore sphere volume = (2/3) × 2πr³ = (4/3)πr³.
This was so remarkable to Archimedes that he requested a sphere-in-cylinder diagram on his tombstone!
Worked Examples
Example 1 – Basketball
A regulation NBA basketball has diameter 24 cm. Find its volume and surface area.
V = (4/3) × π × 12³ = (4/3) × π × 1728 = 2304π ≈ 7238.2 cm³
SA = 4 × π × 144 = 576π ≈ 1809.6 cm²
Example 2 – Water Tank
A spherical water tank has radius 3 m. How many litres can it hold?
Since 1 m³ = 1000 litres: 113,100 litres
Example 3 – Tennis Ball
A tennis ball has diameter 6.7 cm. Find its volume.
V = (4/3)π × 3.35³ = (4/3)π × 37.595 ≈ 157.48 cm³
Sphere vs Cylinder vs Cone Volumes
| Shape | Formula | r=5, h=10 Volume |
|---|---|---|
| Cylinder (r=5, h=10) | πr²h | 785.40 cubic units |
| Sphere (r=5) | (4/3)πr³ | 523.60 cubic units |
| Cone (r=5, h=10) | (1/3)πr²h | 261.80 cubic units |
Notable relationship: Cone : Sphere : Cylinder = 1 : 2 : 3 (when they share the same radius, and the cylinder/cone height = diameter). Archimedes discovered this elegant ratio.
Real-World Applications
- Sports equipment: Balls in football, basketball, soccer, tennis, golf — all spheres of specific volumes
- Storage tanks: Spherical tanks are optimal for storing gas or liquid under pressure (minimum surface area for given volume)
- Astronomy: Calculating planetary volumes and masses (Earth volume ≈ 1.083 × 10¹² km³)
- Medicine: Tumour volume estimation using MRI/CT scan measurements
- Bubbles: Soap bubbles are spherical because a sphere minimizes surface area for a given volume — a principle called the isoperimetric inequality