Volume & Surface Area Calculator
Volume: - cubic units
Surface Area: - square units
Understanding Volume & Surface Area
Basic Concepts
Volume Formulas:
- Cube: V = s³
- Sphere: V = (4/3)πr³
- Cylinder: V = πr²h
- Cone: V = (1/3)πr²h
- Square Pyramid: V = (1/3)a²h
- Rectangular Prism: V = l×w×h
Surface Area Formulas:
- Cube: SA = 6s²
- Sphere: SA = 4πr²
- Cylinder: SA = 2πr² + 2πrh
- Cone: SA = πr² + πr√(r² + h²)
- Square Pyramid: SA = a² + 2a√((a/2)² + h²)
- Rectangular Prism: SA = 2(lw + lh + wh)
Advanced Properties
Similarity
- Volume ratio = (scale factor)³
- Surface area ratio = (scale factor)²
- Linear dimensions ratio = scale factor
- Shape preservation under scaling
Optimization
- Sphere: maximum volume for given surface area
- Cube: minimum surface area for given volume
- Isoperimetric inequality
- Principle of least action
Integration Methods
- Shell method
- Washer method
- Cross-section method
- Triple integrals
Applications
Real-world Usage
- Engineering design
- Architecture
- Manufacturing
- Packaging optimization
Scientific Applications
- Fluid dynamics
- Crystallography
- Molecular modeling
- Astronomical calculations
Advanced Topics
- Non-Euclidean geometry
- Differential geometry
- Topology
- Manifolds