Cylinder vs Cone vs Sphere: Volume Comparison
Compare the volume formulas of cylinders, cones, and spheres. Understand the mathematical relationships between these fundamental 3D shapes.
Three Fundamental Shapes
In geometry, cylinders, cones, and spheres are the three most important curved 3D shapes. Understanding their volume formulas reveals beautiful mathematical relationships.
Volume Formulas
| Shape | Formula | Key Variables |
|---|---|---|
| Cylinder | V = π × r² × h | radius, height |
| Cone | V = ⅓ × π × r² × h | radius, height |
| Sphere | V = ⁴⁄₃ × π × r³ | radius only |
The Cylinder-Cone Relationship
Notice something interesting: a cone’s volume is exactly one-third of a cylinder’s volume when they share the same radius and height.
This means you need exactly three cones of water to fill one cylinder of the same dimensions!
Cylinder volume = 3 × Cone volume (same r and h)
The Cylinder-Sphere Relationship
When a sphere fits perfectly inside a cylinder (same radius, height = diameter = 2r):
- Cylinder volume = π × r² × 2r = 2πr³
- Sphere volume = ⁴⁄₃ × πr³ = ⁴⁄₃πr³
The sphere takes up exactly two-thirds of the cylinder’s volume. Archimedes considered this discovery his greatest achievement!
Practical Comparison
For a radius of 5 cm and height of 10 cm:
- Cylinder: π × 25 × 10 = 785.40 cm³
- Cone: ⅓ × π × 25 × 10 = 261.80 cm³
- Sphere (r=5): ⁴⁄₃ × π × 125 = 523.60 cm³
When to Use Each
- Cylinders: Tanks, pipes, cans, columns
- Cones: Funnels, ice cream cones, roofs
- Spheres: Balls, bubbles, planets