What is 3D Volume Formulas: Cube, Sphere & Cylinder?
The mathematical derivations behind the three most common 3D volume formulas — and why the sphere's 4/3πr³ formula requires integration while the cube and cylinder come directly from base × height.
Mathematical Foundation
Cylinder & Sphere Volumes
= Radius of the circular cross-section
= Height (cylinder only) — the length of the prism
= The integration constant — derived by integrating circular disk areas from -r to +r
Laws & Principles
- Cavalieri's Principle: Two 3D shapes have equal volume if every horizontal plane cuts them into cross-sections of equal area. This principle explains why a cylinder and a prism with the same base area and height have the same volume (V = Bh), regardless of shape. It also allows volume calculations by integrating cross-sectional areas along an axis.
- The Sphere Derivation: Unlike prisms (V = base × height), the sphere requires calculus. Archimedes proved in 225 BCE that V_sphere = (2/3) × V_circumscribed_cylinder using geometric arguments. The integral derivation: V = ∫₋ᵣʳ π(r²-x²)dx = π[r²x - x³/3]₋ᵣʳ = (4/3)πr³. Each infinitesimal disk has area π(r²-x²) at height x.
Step-by-Step Example Walkthrough
" Volume of a sphere with radius 5 units. "
- Formula: V = (4/3) × π × r³.
- r³ = 5³ = 125.
- V = (4/3) × 3.14159 × 125 = 4.18879 × 125 = 523.60 cubic units.
Final Result: Sphere radius 5 = 523.60 cubic units (vs. cylinder r=5, h=10 = 785.40 cubic units)